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.
What are Trigonometric Identities?
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
Reciprocal Identities
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
Pythagorean Trigonometric Identities
The Pythagorean trigonometric identities in trigonometry are derived from the Pythagoras theorem. The following are the 3 Pythagorean trig identities.
 sin^{2}θ + cos^{2}θ = 1. This can also be written as 1  sin^{2}θ = cos^{2} θ ⇒ 1  cos^{2}θ = sin^{2}θ
 sec^{2}θ  tan^{2}θ = 1. This can also be written as sec^{2}θ = 1 + tan^{2}θ ⇒ sec^{2}θ  1 = tan^{2}θ
 csc^{2}θ  cot^{2}θ = 1. This can also be written as csc^{2}θ = 1 + cot^{2}θ ⇒ csc^{2}θ  1 = cot^{2}θ
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
Opposite^{2 }+ Adjacent^{2 }= Hypotenuse^{2} ... (1)
Dividing both sides by Hypotenuse^{2}
Opposite^{2}/Hypotenuse^{2} + Adjacent^{2}/Hypotenuse^{2} = Hypotenuse^{2}/Hypotenuse^{2}
By using the definitions of trig ratios, the above equation becomes
 sin^{2}θ + cos^{2}θ = 1
This is one of the Pythagorean identities. In the same way, we can derive two other Pythagorean trigonometric identities.
 tan^{2}θ + 1 = sec^{2}θ (this can be obtained by dividing both sides of (1) by "Adjacent^{2}")
 1 + cot^{2}θ = cosec^{2}θ (this can be obtained by dividing both sides of (1) by "Opposite^{2}")
Complementary and Supplementary Identities
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 cofunction 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
Sum and Difference Identities
The sum and difference identities include the formulas of sin(A+B), cos(AB), tan(A+B), etc.
 sin (A+B) = sin A cos B + cos A sin B
 sin (AB) = sin A cos B  cos A sin B
 cos (A+B) = cos A cos B  sin A sin B
 cos (AB) = cos A cos B + sin A sin B
 tan (A+B) = (tan A + tan B)/(1  tan A tan B)
 tan (AB) = (tan A  tan B)/(1 + tan A tan B)
Periodic Identities
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 abovementioned sum and difference identities.
Double and Half Angles 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 doubleangle identities.
 sin 2θ = 2 sinθ cosθ
 cos 2θ = cos2θ  sin 2θ
= 2 cos^{2}θ  1
= 1  2sin^{2}θ  tan 2θ = (2tanθ)/(1  tan^{2}θ)
Half Angle Formulas: Using one of the above doubleangle formulas,
cos 2θ = 1  2 sin^{2}θ
2 sin^{2}θ = 1 cos 2θ
sin^{2}θ = (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 halfangle formula of sin.
In the same way, we can derive the other halfangle 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
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 doubleangle 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  sin^{2}x) + (1  2sin^{2}x) sin x
= 2 sin x  2 sin^{3}x + sin x  2 sin^{3}x
= 3 sin x  4 sin^{3}x
Just like this, we can derive the other tripleangle formulas as well.
 sin(3x) = 3sin(x)  4sin^{3}(x)
 cos(3x) = 4cos^{3}(x)  3 cos x
 tan(3x) = (3 tan x  tan^{3}x)/(1  3tan^{2}x)
Sum and Product Identities
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
Sine and Cosine Rule
The trigonometric identities that we have learned are derived using rightangled triangles. There are a few other identities that we use in the case of triangles that are not rightangled.
Sine Rule: The sine rule gives the relation between the angles and the corresponding sides of a triangle. For the nonrightangled 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,
 a^{2} = b^{2} + c^{2 } 2bc·cosA
 b^{2} = c^{2} + a^{2}  2ca·cosB
 c^{2} = a^{2} + b^{2}  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 (180θ).
 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 sin^{2}x + cos^{2}x = 1.
 The halfangle formula of tan is obtained by applying the identity tan = sin/cos and then using the halfangle formulas of sin and cos.
☛ Related Topics:
Trigonometry Identities Examples

Example 1: Prove the following identity using the trig identities:
[(sin 3θ + cos 3θ)/(sin θ + cos θ)] + sin θ cos θ = 1
Solution:
We make use of the following identity:
a^{3}+b^{3} = (a+b)(a^{2}ab+b^{2})
We use the Pythagorean identities to prove this identity.
L.H.S. = [(sin 3θ + cos 3θ)(sin θ + cos θ)] + sin θ cos θ
= [(sin θ + cos θ)(sin^{2}θ  sin θ cos θ + cos^{2}θ)(sin θ + cos θ) + sin θ cos θ
= (sin^{2}θ  sin θ cos θ + cos^{2}θ) + sin θ cos θ
= sin^{2}θ + cos^{2}θ
= 1 = R.H.S.
Answer: The given identity is proved.

Example 2: Prove the following identity using the trigonometry identities:
(sin θ + cosec θ)^{2} + (cos θ + sec θ)^{2} = 7 + tan^{2} θ + cot^{2} θ
Solution:
We use the reciprocal identities and Pythagorean identities to prove this identity.
L H S = (sin^{2} θ + cosec^{2} θ + 2 sin θ cosec θ) + (cos^{2} θ + sec^{2} θ + 2 cos θ sec θ)
= sin^{2} θ + cos^{2} θ + cosec^{2}θ + sec^{2}θ + 2 + 2
= 1 + (1 + cot^{2} θ) + (1 + tan^{2} θ) + 2 + 2
= 7 + tan^{2} θ + cot^{2} θ = R.H.S.
Answer: The given identity is proved.

Example 3: Find the exact value of sin 75° using the trigo identities.
Solution:
We know that 75° = 30° + 45°
We apply the sum identity of sin to find the value of sin 75°.
sin (A+B) = sin A cos B + cos A sin B
Substitute A = 30° and B = 45° here on both sides:
sin (30° + 45°) = sin 30° cos 45°+ cos 30° + sin 45°
sin 75° = (1/2)⋅(√2/2) + (√3/2)⋅(√2/2)
sin 75° = (√2 + √6)/4 = (√3 + 1)2√2
Here, the values of sin 30°, cos 45°, cos 30°, and sin 45° can be obtained by using the trigonometric table.
Answer: sin 75°= (√3 + 1)/2√2
FAQs on Trigonometric Identities
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
 sin^{2}θ + cos^{2}θ = 1
 1 + tan^{2}θ = sec^{2}θ
 1 + cot^{2}θ = cosec^{2}θ
What are the 3 Trigonometric Identities?
The three trigonometric identities are given as,
 sin^{2}θ + cos^{2}θ = 1
 1 + tan^{2}θ = sec^{2}θ
 1 + cot^{2}θ = cosec^{2}θ
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 halfangle identities. For the nonrightangled 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
 sin^{2}θ + cos^{2}θ = 1
 sec^{2}θ  tan^{2}θ = sec^{2}θ
 cosec^{2}θ  cot^{2}θ = 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 evenodd identities, and the angle sum and difference identities.
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