Trigonometric Identities

Trigonometric identities are a fundamental aspect of trigonometry, which is the study of the relationships between the angles and sides of triangles. These identities are mathematical equations that involve trigonometric functions, such as sine, cosine, and tangent, and are true for all values of the variables involved.

Trigonometry identities are useful for simplifying expressions, solving equations, and proving mathematical theorems in various fields of science and engineering. Understanding the properties and applications of these identities is essential for students and professionals in fields such as mathematics, physics, and engineering.

Trigonometric identities are the equations involving trigonometric functions and hold true for every value of the variables involved, provided that both sides of the equality are defined. These are equations are true for any value of the variable that is there in the domain.

The trig identities relate the 6 trigonometric functions sine, cosine, tangent, cosecant, secant, and cotangent. Let's learn about all trigonometric identities in detail which are mentioned below.

• Reciprocal Identities
• Pythagorean Identities
• Opposite Angle Identities
• Complementary Angle Identities
• Supplementary Angle Identities
• Sum and Difference Identities
• Periodic Identities
• Double Angle and Half Angle Identities
• Triple Angle Identities
• Sum to Product Identities
• Product to Sum Identities
• Sine Law and Cosine Law

We already know that the reciprocals of sine, cosine, and tangent are cosecant, secant, and cotangent respectively.

Thus, the reciprocal identities are given as,

• sin θ = 1/cosec θ (OR) cosec θ = 1/sin θ
• cos θ = 1/sec θ (OR) sec θ = 1/cos θ
• tan θ = 1/cot θ (OR) cot θ = 1/tan θ

The Pythagorean trigonometric identities in trigonometry are derived from the Pythagoras theorem. The following are the 3 Pythagorean trig identities.

• sin²θ + cos²θ = 1. This can also be written as 1 - sin²θ = cos² θ ⇒ 1 - cos²θ = sin²θ
• sec²θ - tan²θ = 1. This can also be written as sec²θ = 1 + tan²θ ⇒ sec²θ - 1 = tan²θ
• csc²θ - cot²θ = 1. This can also be written as csc²θ = 1 + cot²θ ⇒ csc²θ - 1 = cot²θ

Let us see how to prove these identities.

3 Pythagorean Identities Proof

Consider the right angle angled triangle ABC which is right angled at B as below.

Applying the Pythagoras theorem to this triangle, we get

Opposite² + Adjacent² = Hypotenuse² ... (1)

Dividing both sides by Hypotenuse²

Opposite²/Hypotenuse² + Adjacent²/Hypotenuse² = Hypotenuse²/Hypotenuse²

By using the definitions of trig ratios, the above equation becomes

• sin²θ + cos²θ = 1

This is one of the Pythagorean identities. In the same way, we can derive two other Pythagorean trigonometric identities.

• tan²θ + 1 = sec²θ (this can be obtained by dividing both sides of (1) by "Adjacent²")
• 1 + cot²θ = cosec²θ (this can be obtained by dividing both sides of (1) by "Opposite²")

The complementary angles are a pair of two angles such that their sum is equal to 90°. The complement of an angle θ is (90 - θ). The trigonometric ratios of complementary angles (also known as co-function Identities) are:

• sin (90°- θ) = cos θ
• cos (90°- θ) = sin θ
• tan (90°- θ) = cot θ
• cosec (90°- θ) = sec θ
• sec (90°- θ) = cosec θ
• cot (90°- θ) = tan θ

The supplementary angles are a pair of two angles such that their sum is equal to 180°. The supplement of an angle θ is (180 - θ). The trigonometric ratios of supplementary angles are:

• sin (180°- θ) = sinθ
• cos (180°- θ) = -cos θ
• tan (180°- θ) = -tan θ
• cosec (180°- θ) = cosec θ
• sec (180°- θ)= -sec θ
• cot (180°- θ) = -cot θ

The sum and difference identities include the formulas of sin(A+B), cos(A-B), tan(A+B), etc.

• sin (A+B) = sin A cos B + cos A sin B
• sin (A-B) = sin A cos B - cos A sin B
• cos (A+B) = cos A cos B - sin A sin B
• cos (A-B) = cos A cos B + sin A sin B
• tan (A+B) = (tan A + tan B)/(1 - tan A tan B)
• tan (A-B) = (tan A - tan B)/(1 + tan A tan B)

Periodic identities in trigonometry are a set of identities that describe the periodic nature of trigonometric functions. A periodic function is a function that repeats its values after a certain interval, known as its period. Here are the periodic identities of sin, cos, and tan.

• sin(x + 2π) = sin(x)
• cos(x + 2π) = cos(x)
• tan(x + π) = tan(x)

We can try deriving these either by the unit circle or the above-mentioned sum and difference identities.

Double angle formulas: The double angle trigonometric identities can be obtained by using the sum and difference formulas.

For example, from the above formulas:

sin (A+B) = sin A cos B + cos A sin B

Substitute A = B = θ on both sides here, we get:

sin (θ + θ) = sinθ cosθ + cosθ sinθ
sin 2θ = 2 sinθ cosθ

In the same way, we can derive the other double-angle identities.

• sin 2θ = 2 sinθ cosθ
• cos 2θ = cos2θ - sin 2θ
  = 2 cos²θ - 1
  = 1 - 2sin²θ
• tan 2θ = (2tanθ)/(1 - tan²θ)

Half Angle Formulas: Using one of the above double-angle formulas,

cos 2θ = 1 - 2 sin²θ

2 sin²θ = 1- cos 2θ

sin²θ = (1 - cos2θ)/(2)

sin θ = ±√[(1 - cos 2θ)/2]

Replacing θ by θ/2 on both sides,

sin (θ/2) = ±√[(1 - cos θ)/2]

This is the half-angle formula of sin.

In the same way, we can derive the other half-angle formulas.

• sin (θ/2) = ±√[(1 - cos θ)/2]
• cos (θ/2) = ±√(1 + cos θ)/2
• tan (θ/2) = ±√[(1 - cos θ)(1 + cos θ)]

Triple angle identities are trigonometric identities that relate the values of trigonometric functions of three times an angle to the values of trigonometric functions of the angle itself. The triple angle formula of sine can be derived in the following way.

We can write sin 3x as:

sin (3x) = sin (2x + x)

= sin 2x cos x + cos 2x sin x

By using the double-angle formula of sine,

sin 3x = 2 sin x cos x cos x + cos 2x sin x

Now, by using the Pythagorean identity and double angle formula of cos,

= 2 sin x (1 - sin²x) + (1 - 2sin²x) sin x

= 2 sin x - 2 sin³x + sin x - 2 sin³x

= 3 sin x - 4 sin³x

Just like this, we can derive the other triple-angle formulas as well.

• sin(3x) = 3sin(x) - 4sin³(x)
• cos(3x) = 4cos³(x) - 3 cos x
• tan(3x) = (3 tan x - tan³x)/(1 - 3tan²x)

These identities are used either to convert "sum into the product" or "product into sum" in the case of trigonometric functions.

Sum to Product Identities: These identities are

• sin A + sin B = 2[sin((A + B)/2)cos((A − B)/2)]
• sin A − sin B = 2[cos((A + B)/2)sin((A − B)/2)]
• cos A + cos B = 2[cos((A + B)/2)cos((A − B)/2)]
• cos A − cos B = −2[sin((A + B)/2)sin((A − B)/2)]

Product to Sum Identities: These identities are:

• sin A⋅cos B = [sin(A + B) + sin(A − B)]/2
• cos A⋅cos B = [cos(A + B) + cos(A − B)]/2
• sin A⋅sin B = [cos(A − B) − cos(A + B)]/2

The trigonometric identities that we have learned are derived using right-angled triangles. There are a few other identities that we use in the case of triangles that are not right-angled.

Sine Rule: The sine rule gives the relation between the angles and the corresponding sides of a triangle. For the non-right-angled triangles, we will have to use the sine rule and the cosine rule. For a triangle with sides 'a', 'b', and 'c' and the respective opposite angles are A, B, and C, the sine rule can be given as,

• a/sinA = b/sinB = c/sinC
• sinA/a = sinB/b = sinC/c
• a/b = sinA/sinB; a/c = sinA/sinC; b/c = sinB/sinC

Cosine Rule: The cosine rule gives the relation between the angles and the sides of a triangle and is usually used when two sides and the included angle of a triangle are given. Cosine rule for a triangle with sides 'a', 'b', and 'c' and the respective opposite angles are A, B, and C, sine rule can be given as,

• a² = b² + c² - 2bc·cosA
• b² = c² + a² - 2ca·cosB
• c² = a² + b² - 2ab·cosC

Important Notes on Trigonometry Identities:

• To write the trigonometric ratios of complementary angles, we consider the following as pairs: (sin, cos), (cosec, sec), and (tan, cot).
• While writing the trigonometric ratios of supplementary angles, the trigonometric ratio won't change. The sign can be decided using the fact that only sin and cosec are positive in the second quadrant where the angle is of the form (180-θ).
• There are 3 formulas for the cos 2x formula. You can remember just the first one because the other two can be obtained by the Pythagorean identity sin²x + cos²x = 1.
• The half-angle formula of tan is obtained by applying the identity tan = sin/cos and then using the half-angle formulas of sin and cos.

☛ Related Topics:

• Trigonometric Identities for Class 12
• Trigonometric Identities Class 11
• Trigonometry Identities Class 10

What are Trigonometry Identities in Trigonometry?

Trigonometric identities are the equalities involving trigonometric functions and hold true for every value of the variables involved, in a manner that both sides of the equality are defined. Some important identities in trigonometry are given as,

• sin θ = 1/cosec θ
• cos θ = 1/sec θ
• tan θ = 1/cot θ
• sin²θ + cos²θ = 1
• 1 + tan²θ = sec²θ
• 1 + cot²θ = cosec²θ

What are the 3 Trigonometric Identities?

The three trigonometric identities are given as,

• sin²θ + cos²θ = 1
• 1 + tan²θ = sec²θ
• 1 + cot²θ = cosec²θ

What are Trigonometry Identities used for?

The main uses of trig identities include:

• Simplifying expressions: Trigonometric identities allow us to simplify complex trigonometric expressions by replacing them with the simpler form which can be useful in solving trigonometric equations, graphing trigonometric functions, and simplifying calculations.
• Solving equations: Trigonometric identities can be used to solve trigonometric equations by transforming them into simpler forms. This is often done by using more than one identity to reduce the equation to a form that can be solved more easily.
• Proving theorems: Trigonometric identities are often used in the proof of mathematical theorems, particularly in geometry and calculus. By using them, mathematicians can establish the validity of various geometric and calculus formulas.
• Calculating values: Trigonometric identities are used to calculate the values of trigonometric functions, such as sine, cosine, and tangent, for various angles. This is important in many applications, such as in navigation, surveying, and engineering.

How to Prove Trigonometric Identities?

Trigonometric Identities can be proved by using other known Pythagorean and trigonometric identities. We can use some trigonometric ratios and formulas as well to prove the trig identities.

What are Opposite Angle Identities in Trigonometry?

The opposite angle identities talk about what happens to trig ratios when the angle is negative. They are as follows:

• sin(-x) = - sin x; csc(-x) = - csc x
• cos(-x) = cos x; sec(-x) = sec x
• tan(-x) = - tan x; cot(-x) = - cot x

These identities can be derived from the definitions of trigonometric functions and the properties of the unit circle.

How Do you Solve equations with Trig Identities?

Various math problems can be solved using trigonometric identities. We can convert equations into Parametric equations and then apply the trigonometric identities to solve them.

What Trigonometric Identities We Must Know?

All trig identities are used in solving the problems. The main trigonometric identities are Pythagorean identities, reciprocal identities, sum and difference identities, and double angle and half-angle identities. For the non-right-angled triangles, we will have to use the sine rule and the cosine rule.

What are Eight Fundamental Trigonometric Identities?

The eight fundamental trigonometric identities are:

• cosec θ = 1/sin θ
• sec θ = 1/cos θ
• cot θ = 1/tan θ
• sin²θ + cos²θ = 1
• sec²θ - tan²θ = sec²θ
• cosec²θ - cot²θ = 1
• tanθ = sinθ/cos θ
• cot θ = cosθ/sinθ

What are Some Common Trigonometric Identities?

Some common trigonometric identities include the Pythagorean identities, the reciprocal identities, the quotient identity, the even-odd identities, and the angle sum and difference identities.