Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values of the variables within their domains. 

There are many such identities, either involving the sides of a right-angled triangle, its angle, or both. They are based on the six fundamental trigonometric functions: sine (sin), cosine (cos), tangent (tan), cosecant (cosec), secant (sec), and cotangent (cot).  

All the identities are derived from the six trigonometric functions and are used to simplify expressions, verify equations, and solve trigonometric problems. 

List

Trigonometric identities are classified based on the type of relationships they describe among trigonometric functions.

Reciprocal Identities

These identities express the reciprocal relationships between sine, cosine, tangent, and their corresponding co-functions.

• ${\sin \theta =\dfrac{1}{\text{cosec}\, \theta }}$
• ${\text{cosec}\, \theta =\dfrac{1}{\sin \theta }}$

• ${\cos \theta =\dfrac{1}{\sec \theta }}$
• ${\sec \theta =\dfrac{1}{\cos \theta }}$

• ${\tan \theta =\dfrac{1}{\cot \theta }}$
• ${\cot \theta =\dfrac{1}{\tan \theta }}$

Pythagorean Identities

There are 3 Pythagorean identities in trigonometry, and they are based on the right-angled triangle rule or Pythagorean theorem.

• sin² θ + cos² θ = 1
• 1 + tan² θ = sec² θ
• 1 + cot² θ = cosec² θ

Proof of sin² θ + cos² θ = 1

Let us consider a right-angled triangle ABC, which is right-angled at C, as shown.

By applying the Pythagoras theorem to this triangle, we get

Opposite² + Adjacent² = Hypotenuse² …..(i)

On dividing both sides of equation (i) by Hypotenuse², we get

${\dfrac{Opposite^{2}}{Hypotenuse^{2}}+\dfrac{Adjacent^{2}}{Hypotenuse^{2}}=\dfrac{Hypotenuse^{2}}{Hypotenuse^{2}}}$

From the definitions of the trigonometric ratios,

⇒ sin² θ + cos² θ = 1

Proof of 1 + tan² θ = sec² θ

As we know from the right-angled triangle rule,

Opposite² + Adjacent² = Hypotenuse² …..(i)

Dividing both sides of equation (i) by Adjacent², we get

${\dfrac{Opposite^{2}}{Adjacent^{2}}+\dfrac{Adjacent^{2}}{Adjacent^{2}}=\dfrac{Hypotenuse^{2}}{Adjacent^{2}}}$

From the definitions of the trigonometric ratios,

⇒ 1 + tan² θ = sec² θ

Proof of 1 + cot² θ = cosec² θ

Similarly, 

Dividing both sides of equation (i) by Opposite², we get

${\dfrac{Opposite^{2}}{Opposite^{2}}+\dfrac{Adjacent^{2}}{Opposite^{2}}=\dfrac{Hypotenuse^{2}}{Opposite^{2}}}$

By using the definitions of the trigonometric ratios,

⇒ 1 + cot² θ = cosec² θ

Quotient Identities

These identities define tangent and cotangent in terms of sine and cosine:

• tan θ = ${\dfrac{\sin \theta }{\cos \theta }}$
• cot θ = ${\dfrac{\cos \theta }{\sin \theta }}$

Complementary Angles Identities

Complementary identities (also known as co-function identities) show how the function of an angle θ relates to the co-function of its complement angle (90° – θ):

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

Supplementary Angles Identities

These identities represent how the trigonometric values of an angle θ are related to its supplementary angle (180° – θ):

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

Opposite Angles Identities

Even-odd identities describe the behavior of trigonometric functions for opposite angles (−θ) and highlight their symmetry properties. These identities classify trigonometric functions as either even or odd based on how their values change with the sign of the angle.

• sin (-θ) = -sin θ (odd function)
• cos (-θ) = cos θ (even function)
• tan (-θ) = – tan θ (odd function)
• cot (-θ) = -cot θ (odd function)
• sec (-θ) = sec θ (even function)
• cosec (-θ) = -cosec θ (odd function)

Periodic Identities

Trigonometric functions are periodic, which means they repeat themselves after a regular interval, called periods. Each function has a different period. 

• sin (2nπ + θ) = sin θ
• cos (2nπ + θ) = cos θ
• tan (nπ + θ) = tan θ
• cot (nπ + θ) = cot θ
• sec (2nπ + θ) = sec θ
• cosec (2nπ + θ) = cosec θ

Here, n ∈ ℤ (set of all integers)

Note: The sine, cosine, and their reciprocals (cosecant and secant) functions have a period of 2π or 360°. Moreover, the tangent and cotangent functions have a period of π or 180°.

Sum and Difference of Angles Identities

The identities involving the sum or difference of two angles for the sine, cosine, and tangent functions are as follows:

• 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) = ${\dfrac{\tan A+\tan B}{1-\tan A\tan B}}$
• tan (A – B) = ${\dfrac{\tan A-\tan B}{1+\tan A\tan B}}$ 

Double Angle Identities

If the angles of the six trigonometric functions are doubled, then the identities are:

• sin 2θ = 2sin θ ⋅ cos θ = ${\dfrac{2\tan \theta }{1-\tan ^{2}\theta }}$
• cos 2θ = cos² θ – sin² θ = ${\dfrac{1-\tan ^{2}\theta }{1+\tan ^{2}\theta }}$
• cos 2θ = 2cos² θ – 1 = 1 – 2 sin² θ
• tan 2θ = ${\dfrac{2\tan \theta }{1+\tan ^{2}\theta }}$
• sec 2θ = ${\dfrac{\sec ^{2}\theta }{2-\sec ^{2}\theta }}$
• cosec 2θ = ${\dfrac{\sec \theta \cdot \text{cosec}\, \theta }{2}}$
• cot 2θ = ${\dfrac{\cot ^{2}\theta -1}{2\cot \theta }}$

Half-Angle Identities

If the angles of the six trigonometric functions are halved, then the identities are:

• ${\sin \dfrac{\theta }{2}=\pm \sqrt{\dfrac{1-\cos \theta }{2}}}$
• ${\cos \dfrac{\theta }{2}=\pm \sqrt{\dfrac{1+\cos \theta }{2}}}$
• ${\tan \dfrac{\theta }{2}=\pm \sqrt{\dfrac{1-\cos \theta }{1+\cos \theta }}}$

Triple Angle Identities

When the angle of a trigonometric function is three times a given angle (3θ), the corresponding identities are:

• sin 3θ = 3sin θ – 4sin³ θ
• cos 3θ = 4cos³ θ – 3cos θ
• tan 3θ = ${\dfrac{3\tan \theta -\tan ^{3}\theta }{1-3\tan ^{2}\theta }}$

Product-to-Sum Identities

The product-to-sum identities express products of trigonometric functions as their sums or differences:

• 2sin A ⋅ cos B = sin(A + B) + sin(A – B)
• 2cos A ⋅ cos B = cos(A + B) + cos(A – B)
• 2sin A ⋅ sin B = cos(A – B) – cos(A + B)

Sum-to-Product Identities

The sum-to-product identities express sums of trigonometric functions as products of those functions:

• ${\sin A+\sin B=2\sin \left( \dfrac{A+B}{2}\right) \cos \left( \dfrac{A-B}{2}\right)}$
• ${\sin A-\sin B=2\sin \left( \dfrac{A-B}{2}\right) \cos \left( \dfrac{A+B}{2}\right)}$
• ${\cos A+\cos B=-2\sin \left( \dfrac{A+B}{2}\right) \sin \left( \dfrac{A-B}{2}\right)}$
• ${\cos A-\cos B=2\cos \left( \dfrac{A+B}{2}\right) \cos \left( \dfrac{A-B}{2}\right)}$

Triangle Identities (Sine, Cosine, Tangent rule)

These identities are applicable for all the triangles, not just for the right triangles. 

If A, B, and C are the vertices of a given triangle and a, b, and c are its corresponding sides, then:

Sine Rule 

• ${\dfrac{\sin A}{a}=\dfrac{\sin B}{b}=\dfrac{\sin C}{C}}$ 

Or 

• ${\dfrac{a}{\sin A}=\dfrac{b}{\sin B}=\dfrac{C}{\sin C}}$

Cosine Rule

• ${\cos A=\dfrac{b^{2}+c^{2}-a^{2}}{2bc}}$

Or

• ${\cos B=\dfrac{a^{2}+c^{2}-b^{2}}{2ac}}$

Or

• ${\cos C=\dfrac{a^{2}+b^{2}-c^{2}}{2ab}}$

Tangent Rule

• ${\dfrac{a-b}{a+b}=\dfrac{\tan \left( \dfrac{A-B}{2}\right) }{\tan \left( \dfrac{A+B}{2}\right) }}$

Or

• ${\dfrac{b-c}{b+c}=\dfrac{\tan \left( \dfrac{B-C}{2}\right) }{\tan \left( \dfrac{B+C}{2}\right) }}$

Or

• ${\dfrac{c-a}{c+a}=\dfrac{\tan \left( \dfrac{C-A}{2}\right) }{\tan \left( \dfrac{C+A}{2}\right) }}$

Solved Examples