Trigonometry Formulas and Identities: Complete List

Trigonometry Formulas – List of All Trigonometric Identities and Formulas

Trigonometry is a branch of mathematics that studies the relationships between angles and the sides of triangles, especially right triangles. It plays a key role in various fields such as physics, engineering, astronomy, and computer graphics. Understanding trigonometry formulas and identities is essential for solving a wide range of problems in these domains.

Also Read:-

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Numbers in Words: Number Names and Examples from 1 to 1000

Here are some important types of trigonometry formulas and identities:

1. Basic Trigonometric Functions
The primary trigonometric functions are defined using a right triangle:

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Sine (sin): $sin (θ) = hypotenuse opposite$
Cosine (cos): $cos (θ) = hypotenuse adjacent$
Tangent (tan): $tan (θ) = c o s ( θ ) s i n ( θ ) = adjacent opposite$

2. Pythagorean Theorem
The Pythagorean theorem is fundamental to trigonometry:

$a^{2} + b^{2} = c^{2}$
Where $ a $ and $ b $ are the lengths of the legs of a right triangle, and $ c $ is the length of the hypotenuse.

3. Pythagorean Identities
These identities are derived from the Pythagorean theorem and involve the square of the trigonometric functions:

$sin^{2} (θ) + cos^{2} (θ) = 1$
$1 + tan^{2} (θ) = sec^{2} (θ)$
$1 + cot^{2} (θ) = csc^{2} (θ)$

4. Reciprocal Identities
These express one trigonometric function in terms of another:

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$sec (θ) = c o s ( θ ) 1$
$csc (θ) = s i n ( θ ) 1$
$cot (θ) = t a n ( θ ) 1$

5. Angle Sum and Difference Identities
These formulas relate the trigonometric functions of sums and differences of angles:

Sine: $sin (α + β) = sin (α) cos (β) + cos (α) sin (β)$
Cosine: $cos (α + β) = cos (α) cos (β) - sin (α) sin (β)$
Tangent: $tan (α + β) = 1 - t a n ( α ) t a n ( β ) t a n ( α ) + t a n ( β )$

6. Double Angle Identities
These identities express trigonometric functions at double angles:

Sine: $sin (2 θ) = 2 sin (θ) cos (θ)$
Cosine: $cos (2 θ) = cos^{2} (θ) - sin^{2} (θ)$
Tangent: $tan (2 θ) = 1 - t a n ^{2} ( θ ) 2 t a n ( θ )$

7. Half Angle Identities
These identities express trigonometric functions at half angles:

Sine: $sin (2 θ) = \pm 2 1 - c o s ( θ )$
Cosine: $cos (2 θ) = \pm 2 1 + c o s ( θ )$
Tangent: $tan (2 θ) = \pm 1 + c o s ( θ ) 1 - c o s ( θ )$

8. Law of Sines and Law of Cosines
These laws relate the sides and angles of any triangle, not just right triangles:

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Law of Sines: $a s i n ( A ) = b s i n ( B ) = c s i n ( C )$
Law of Cosines: $c^{2} = a^{2} + b^{2} - 2 ab cos (C)$

These laws are essential for solving non-right triangles.

9. Unit Circle
The unit circle is a graphical representation of trigonometric functions. It is a circle with a radius of one, centered at the origin of a coordinate system. Trigonometric functions can be derived from the unit circle:

– The angle in radians around the circle is measured counterclockwise.
– Points on the circle represent the sine and cosine values for angles from 0 to $2\pi$.

Understanding these formulas and identities helps you solve trigonometric problems and equations efficiently. Practice problems are useful for reinforcing your knowledge and improving your skills.

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Trigonometry Formulas

What is Trigonometry?

Trigonometry is a branch of mathematics that focuses on studying the relationships between angles and lengths in triangles. It originated from geometric problems involving right triangles but has since expanded to encompass various shapes and periodic phenomena. Trigonometry is built on the fundamental trigonometric functions—sine, cosine, and tangent—which describe the ratios of certain side lengths in right triangles to their angles.

These functions form the basis for understanding and solving problems involving angles, distances, and heights in various fields such as physics, engineering, architecture, and astronomy. For example, trigonometry is used to calculate the height of buildings, the distance between celestial objects, and the behavior of waves.

Additionally, trigonometry plays a crucial role in understanding periodic functions and patterns, as well as in the analysis of sound and light waves. Trigonometric identities and formulas, such as the Pythagorean identity, are essential tools for simplifying and solving complex mathematical problems. Overall, trigonometry is a versatile and essential field of mathematics that underpins many scientific and engineering applications.

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Angles (In Degrees)	0°	30°	45°	60°	90°	180°	270°	360°
Angles (In Radians)	0°	π/6	π/4	π/3	π/2	π	3π/2	2π
sin	0	1/2	1/√2	√3/2	1	0	-1	0
cos	1	√3/2	1/√2	1/2	0	-1	0	1
tan	0	1/√3	1	√3	∞	0	∞	0
cot	∞	√3	1	1/√3	0	∞	0	∞
cosec	∞	2	√2	2/√3	1	∞	-1	∞
sec	1	2/√3	√2	2	∞	-1	∞	1
Trigonometry Ratios Table

Trigonometry Functions

Trigonometric functions are mathematical tools that relate angles in right triangles to the lengths of the sides. These functions have wide-ranging applications across physics, engineering, astronomy, and other scientific fields.

The primary trigonometric functions include:

Sine (sin): Relates the length of the side opposite an angle to the length of the hypotenuse in a right triangle.
Cosine (cos): Relates the length of the adjacent side to the length of the hypotenuse.
Tangent (tan): Relates the length of the side opposite an angle to the length of the adjacent side.

The secondary functions include:

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Cotangent (cot): The reciprocal of tangent.
Secant (sec): The reciprocal of cosine.
Cosecant (csc): The reciprocal of sine.

These functions are essential for solving problems involving angles, distances, and waveforms. They also help in modeling periodic phenomena and analyzing sound and light waves. Trigonometric functions are foundational in various mathematical and practical applications.

Trigonometric Function	Domain	Range	Period
sin(θ)	All Real Number i.e., R	[-1, 1]	2π or 360°
cos(θ)	All Real Numbers i.e.,	[-1, 1]	2π or 360°
tan(θ)	All Real Numbers excluding odd multiples of π/2	R	π or 180°
cot(θ)	All Real Numbers excluding multiples of π	R	2π or 360°
sec(θ)	All Real Numbers excluding values where cos(x) = 0	R-[-1, 1]	2π or 360°
cosec(θ)	All Real Numbers excluding multiples of π	R-[-1, 1]	π or 180°

Trigonometry Formula Overview

Trigonometry is a branch of mathematics that studies the relationships between angles and the sides of triangles, particularly right-angled triangles. In a right-angled triangle, one of the angles is a right angle (90 degrees), which gives rise to specific relationships between the three sides of the triangle.

The three sides of a right-angled triangle are:

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Hypotenuse: This is the longest side of the triangle and is opposite the right angle. It represents the diagonal distance between the two other sides.
Opposite Side (Perpendicular): This side is opposite to the angle in question (not the right angle). It forms the other part of the right angle.
Adjacent Side (Base): This side is adjacent to the angle in question and is not the hypotenuse. It forms the base of the triangle along with the hypotenuse.

These sides and their relationships with the angles in the triangle give rise to the fundamental trigonometric functions: sine, cosine, and tangent.

Sine (sin): The ratio of the length of the opposite side to the length of the hypotenuse. $sin (θ) = hypotenuse opposite side$
Cosine (cos): The ratio of the length of the adjacent side to the length of the hypotenuse. $cos (θ) = hypotenuse adjacent side$
Tangent (tan): The ratio of the length of the opposite side to the length of the adjacent side. $tan (θ) = adjacent side opposite side$

The inverse functions (arcsine, arccosine, and arctangent) can be used to find the angles when the ratios are known.

Understanding the relationships between the sides and angles of a right-angled triangle is fundamental to trigonometry. These principles can be extended to other geometric shapes and contexts, providing the basis for more advanced mathematical, engineering, and physical applications.

Trigonometry Formulas

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Trigonometry Ratio

Trigonometry is a branch of mathematics that studies the relationships between angles and the sides of triangles, particularly right-angled triangles. This area of mathematics provides a wide range of formulas and identities that are critical for solving problems in various mathematical and scientific fields. Below, you’ll find an overview of the different types of trigonometry formulas and ratios that are essential for students in Classes 9, 10, 11, and 12.

Here is the list of formulas in trigonometry we are going to discuss:

Basic Trigonometric Ratio Formulas
Unit Circle Formulas
Trigonometric Identities

Basic Trigonometric Ratios

In trigonometry, there are six fundamental functions known as trigonometric ratios. These ratios describe the relationships between the angles and sides of a right triangle. The six trigonometric functions are:

Sine (sin): The ratio of the length of the side opposite an angle to the length of the hypotenuse.
Cosine (cos): The ratio of the length of the adjacent side to the length of the hypotenuse.
Tangent (tan): The ratio of the length of the side opposite an angle to the length of the adjacent side.
Cotangent (cot): The reciprocal of tangent, it relates the length of the adjacent side to the length of the side opposite an angle.
Secant (sec): The reciprocal of cosine, it relates the length of the hypotenuse to the length of the adjacent side.
Cosecant (csc): The reciprocal of sine, it relates the length of the hypotenuse to the length of the side opposite an angle.

These trigonometric ratios are essential in solving problems involving angles, lengths, and periodic functions, and they have extensive applications in various fields such as physics, engineering, and mathematics.

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List of Trigonometric Ratios
Trigonometric Ratio	Definition
sin θ	Perpendicular / Hypotenuse
cos θ	Base / Hypotenuse
tan θ	Perpendicular / Base
sec θ	Hypotenuse / Base
cosec θ	Hypotenuse / Perpendicular
cot θ	Base / Perpendicular

Unit Circle Formula in Trigonometry

The unit circle is a fundamental concept in trigonometry that serves as the basis for understanding and calculating trigonometric functions for all angles. The unit circle is a circle with a radius of 1, centered at the origin of a coordinate system. Here’s how trigonometric functions are derived from the unit circle and how they are related to the coordinates of a point on the circle.

In the unit circle, an angle $θ$ is measured counterclockwise from the positive x-axis. As the angle increases, the point on the unit circle corresponding to the angle moves along the circle.

Given a point $(x, y)$ on the unit circle, where the x-coordinate represents the horizontal distance from the origin and the y-coordinate represents the vertical distance from the origin, the following trigonometric functions can be defined:

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Sine (sin): The sine of angle $θ$ is equal to the y-coordinate of the point on the unit circle corresponding to angle $θ$ . $sin (θ) = y$
Cosine (cos): The cosine of angle $θ$ is equal to the x-coordinate of the point on the unit circle corresponding to angle $θ$ . $cos (θ) = x$
Tangent (tan): The tangent of angle $θ$ is the ratio of the sine of $θ$ to the cosine of $θ$ . It can also be interpreted as the slope of the line from the origin to the point on the unit circle. $tan (θ) = x y$
Cotangent (cot): The cotangent of angle $θ$ is the reciprocal of the tangent of $θ$ . $cot (θ) = y x$
Secant (sec): The secant of angle $θ$ is the reciprocal of the cosine of $θ$ . $sec (θ) = x 1$
Cosecant (csc): The cosecant of angle $θ$ is the reciprocal of the sine of $θ$ . $csc (θ) = y 1$

These functions and their relationships form the basis of trigonometry and can be used to calculate the sides and angles of right-angled triangles. The unit circle provides a convenient way to visualize and understand the behavior of trigonometric functions across different angles and quadrants.

Trigonometric Identities

Trigonometric identities are equations that express relationships between trigonometric functions. They are universally true for all values of the variables within their domains. These identities are essential for simplifying complex expressions, solving equations, and understanding the behavior of trigonometric functions.

Reciprocal Identities
Pythagorean Identities
Periodicity Identities (in Radians)
Even and Odd Angle Formula
Cofunction identities (in Degrees)
Sum and Difference Identities
Double Angle Identities
Inverse Trigonometry Formulas
Triple Angle Identities
Half Angle Identities
Sum to Product Identities
Product Identities

Let’s discuss these identites in detail.

Reciprocal Identities

eciprocal identities express each of the primary trigonometric functions in terms of its reciprocal. They are derived from the definition of the basic trigonometric functions and provide alternate forms for the functions:

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Cosecant (csc): The reciprocal of sine. $csc (θ) = s i n ( θ ) 1$
Secant (sec): The reciprocal of cosine. $sec (θ) = c o s ( θ ) 1$
Cotangent (cot): The reciprocal of tangent. $cot (θ) = t a n ( θ ) 1$

Likewise, the reciprocal identities can also be expressed in terms of the primary functions:

$sin (θ) = c s c ( θ ) 1$
$cos (θ) = s e c ( θ ) 1$
$tan (θ) = c o t ( θ ) 1$

Pythagorean Identities

Pythagorean identities are derived from the Pythagorean theorem and express relationships between the squares of trigonometric functions:

$sin^{2} (θ) + cos^{2} (θ) = 1$
$1 + tan^{2} (θ) = sec^{2} (θ)$
$1 + cot^{2} (θ) = csc^{2} (θ)$

These identities are useful for converting between trigonometric functions and simplifying complex expressions.

Periodicity Identities (in Radians)

Periodicity identities express the cyclic nature of trigonometric functions. These identities indicate how trigonometric functions repeat their values after a certain interval:

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Trigonometric Functions in Four Quadrants

All trigonometric identities repeat themselves after a particular period. Hence are cyclic in nature. This period for the repetition of values is different for different trigonometric identities.

sin (π/2 – A) = cos A & cos (π/2 – A) = sin A

sin (π/2 + A) = cos A & cos (π/2 + A) = – sin A

sin (3π/2 – A) = – cos A & cos (3π/2 – A) = – sin A

sin (3π/2 + A) = – cos A & cos (3π/2 + A) = sin A

sin (π – A) = sin A & cos (π – A) = – cos A

sin (π + A) = – sin A & cos (π + A) = – cos A

sin (2π – A) = – sin A & cos (2π – A) = cos A

sin (2π + A) = sin A & cos (2π + A) = cos A

Here’s a table that compares the trigonometric properties in different quadrants :

Quadrant	Sine (sin θ)	Cosine (cos θ)	Tangent (tan θ)	Cosecant (csc θ)	Secant (sec θ)	Cotangent (cot θ)
I (0° to 90°)	Positive	Positive	Positive	Positive	Positive	Positive
II (90° to 180°)	Positive	Negative	Negative	Positive	Negative	Negative
III (180° to 270°)	Negative	Negative	Positive	Negative	Negative	Positive
IV (270° to 360°)	Negative	Positive	Negative	Negative	Positive	Negative

Even and Odd Angle Formula

Trigonometric functions can be classified as even or odd, depending on whether their signs change with the sign of the angle:

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Even Function: Cosine and secant are even functions, which means they remain unchanged when the angle is negated.
- $cos (- θ) = cos (θ)$
- $sec (- θ) = sec (θ)$
Odd Function: Sine, cosecant, tangent, and cotangent are odd functions, which means their signs reverse when the angle is negated.
- $sin (- θ) = - sin (θ)$
- $csc (- θ) = - csc (θ)$
- $tan (- θ) = - tan (θ)$
- $cot (- θ) = - cot (θ)$

These formulas are useful for simplifying trigonometric expressions and solving equations involving negative angles.

Cofunction identities (in Degrees)

Cofunction identities express the relationship between trigonometric functions of complementary angles (90° – θ). These identities allow us to relate sine to cosine, tangent to cotangent, and secant to cosecant:

sin(90° - x) = cos x and cos(90° - x) = sin x
tan(90° - x) = cot x and cot(90° - x) = tan x
sec(90° - x) = csc x and cosec(90° - x) = sec x

Sum and Difference Identities

Sum and difference identities relate the sine, cosine, and tangent of the sum or difference of two angles to the trigonometric functions of the individual angles:

sin(x + y) = sin(x)cos(y) + cos(x)sin(y) and sin(x - y) = sin(x)cos(y) - cos(x)sin(y)

cos(x + y) = cos(x)cos(y) - sin(x)sin(y) and cos(x - y) = cos(x)cos(y) + sin(x)sin(y)

tan(x + y) = (tan(x) + tan(y)) / (1 - tan(x)tan(y)) and tan(x - y) = (tan(x) - tan(y)) / (1 + tan(x)tan(y))

Double Angle Identities

Double angle identities express trigonometric functions of double the angle (2θ) in terms of the original angle (θ):

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sin (2x) = 2sin(x) • cos(x) = [2tan x/(1 + tan² x)]

cos (2x) = cos²(x) – sin²(x) = [(1 – tan² x)/(1 + tan² x)] = 2cos²(x) – 1 = 1 – 2sin²(x)

tan (2x) = [2tan(x)]/ [1 – tan²(x)]

sec (2x) = sec²x/(2 – sec² x)

cosec (2x) = (sec x • cosec x)/2

Inverse Trigonometry Formulas

Inverse trigonometry formulas involve the inverses of the basic trigonometric functions and are used to find the angle that corresponds to a given trigonometric ratio:

sin^-1 (–x) = – sin^-1 x

cos^-1 (–x) = π – cos^-1 x

tan^-1 (–x) = – tan^-1 x

cosec^-1 (–x) = – cosec^-1 x

sec^-1 (–x) = π – sec^-1 x

cot^-1 (–x) = π – cot^-1 x

Triple Angle Identities

Triple angle identities express trigonometric functions of triple the angle (3θ) in terms of the original angle (θ):

sin 3x=3sin x – 4sin³x

cos 3x=4cos³x – 3cos x

$tan 3x = 1 - 3 t a n ^{2} x 3 t an x - t a n ^{3} x$

Half Angle Identities

Half angle identities express trigonometric functions of half an angle (θ/2) in terms of the original angle:

$sin 2 x = \pm 2 1 - cos x$

$cos 2 x = \pm 2 1 + cos x$

$tan (2 x) = \pm 1 + cos ( x ) 1 - cos ( x )$

Also,

$tan (2 x) = \pm 1 + cos ( x ) 1 - cos ( x )$

$tan (2 x) = \pm ( 1 + cos ( x )) ( 1 - cos ( x )) ( 1 - cos ( x )) ( 1 - cos ( x ))$

$= 1 - co s ^{2} ( x ) ( 1 - cos ( x ) ) ^{2}$

$= s i n ^{2} ( x ) ( 1 - cos ( x ) ) ^{2}$

$= s in ( x ) 1 - cos ( x )$

$tan (2 x) = s in ( x ) 1 - cos ( x )$

Sum to Product Identities

Sum to product identities express sums or differences of trigonometric functions as products:

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sinx + siny = 2[sin((x + y)/2)cos((x − y)/2)]

sinx − siny = 2[cos((x + y)/2)sin((x − y)/2)]

cosx + cosy = 2[cos((x + y)/2)cos((x − y)/2)]

cosx − cosy = −2[sin((x + y)/2)sin((x − y)/2)]

Product Identities

Product identities express products of trigonometric functions as sums or differences:

These trigonometric formulas are derived from the sum and difference formulas for sine and cosine.

sinx⋅cosy = [sin(x + y) + sin(x − y)]/2

cosx⋅cosy = [cos(x + y) + cos(x − y)]/2

sinx⋅siny = [cos(x − y) − cos(x + y)]/2

List of Trigonometry Formulas

The table given below consists of basic trigonometry ratios for angles such as such as 0°, 30°, 45°, 60°, and 90° that are commonly used for solving problems.

Table of Trigonometric Ratios Advertisements
Angles(In Degrees)	0	30	45	60	90	180	270	360
Angles(In Radians)	0	π/6	π/4	π/3	π/2	π	3π/2	2π
sin	0	1/2	1/√2	√3/2	1	0	-1	0
cos	1	√3/2	1/√2	1/2	0	-1	0	1
tan	0	1/√3	1	√3	∞	0	∞	0
cot	∞	√3	1	1/√3	0	∞	0	∞
cosec	∞	2	√2	2/√3	1	∞	-1	∞
sec	1	2/√3	√2	2	∞	-1	∞	1

Solved Questions on Trigonometry Formula:-

Here are 9 solved questions on trigonometry formulas to help you get a better grasp of the concepts:

Question 1

Given: $tan α + sec α = 2$

Find: $tan α - sec α$

Solution:

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Given $tan α + sec α = 2$

To find $tan α - sec α$ , use the formula:

$(tan α + sec α) \times (tan α - sec α) = tan^{2} α - sec^{2} α$

Since $sec^{2} α - tan^{2} α = 1$ , we have:

$(2) (tan α - sec α) = 1$

So, $tan α - sec α = 2 1$

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Question 2

Given: $sin β = 5 4$

Find: $cos β$

Solution:

Using Pythagorean identity: $sin^{2} β + cos^{2} β = 1$

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Given $sin β = 5 4$ , so:

$(5 4)^{2} + cos^{2} β = 1$
$25 16 + cos^{2} β = 1$
$cos^{2} β = 25 9$

Therefore, $cos β = 5 3$

Question 3

Given: $sin γ = 5 3$

Find: $tan γ$

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Solution:

Given $sin γ = 5 3$

From Pythagorean identity: $cos γ = 1 - sin^{2} γ$

So, $cos γ = 1 - (5 3)^{2}$

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$cos γ = 25 16 = 5 4$

Then, use the definition of $tan γ$ :

$tan γ = c o s γ s i n γ$
$tan γ = 5 3 \div 5 4 = 4 3$

Therefore, $tan γ = 4 3$

Question 4

Given: $cos (2 θ)$

Find: $cos^{2} θ - sin^{2} θ$

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Solution:

Use the double angle formula:

$cos (2 θ) = cos^{2} θ - sin^{2} θ$

Hence, $cos^{2} θ - sin^{2} θ = cos (2 θ)$

Question 5

Given: $sin (4 5^{\circ})$

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Find: $cos (4 5^{\circ})$

Solution:

Both $sin (4 5^{\circ})$ and $cos (4 5^{\circ})$ are equal:

$sin (4 5^{\circ}) = cos (4 5^{\circ}) = 2 2$

Therefore, $cos (4 5^{\circ}) = 2 2$

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Question 6

Given: $cos (3 0^{\circ})$

Find: $tan (3 0^{\circ})$

Solution:

Given $cos (3 0^{\circ}) = 2 3$ and $sin (3 0^{\circ}) = 2 1$ , then:

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$tan (3 0^{\circ}) = c o s ( 30 ^{\circ} ) s i n ( 30 ^{\circ} )$
$tan (3 0^{\circ}) = 2 1 \div 2 3 = 3 1$

Therefore, $tan (3 0^{\circ}) = 3 1$

Question 7

Given: $sin ϕ = cos (9 0^{\circ} - ϕ)$

Prove: $cos ϕ = sin (9 0^{\circ} - ϕ)$

Solution:

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Given $sin ϕ = cos (9 0^{\circ} - ϕ)$

By definition:

$cos (9 0^{\circ} - ϕ) = sin ϕ$
Therefore, $cos ϕ = sin (9 0^{\circ} - ϕ)$

Hence, the statement is proven.

Question 8

Given: $sin^{2} θ + cos^{2} θ = 1$

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Find: $tan^{2} θ + 1 = sec^{2} θ$

Solution:

Using the Pythagorean identity:

$sin^{2} θ + cos^{2} θ = 1$
Divide each side by $cos^{2} θ$ :
$tan^{2} θ + 1 = sec^{2} θ$

Hence, the identity is proven.

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Question 9

Given: $cos^{2} x - sin^{2} x = cos (2 x)$

Find: Prove that $2 sin x cos x = sin (2 x)$

Solution:

Given $cos^{2} x - sin^{2} x = cos (2 x)$ , from the double angle formula, $cos^{2} x - sin^{2} x = cos (2 x)$ .

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Now, prove the other double angle identity:

$2 sin x cos x = sin (2 x)$

Given:

$sin (2 x) = 2 sin x cos x$

Hence, $2 sin x cos x = sin (2 x)$ .