Trigonometric Identities (List of Trigonometric Identities | Proofs | PDFs) (2024)

Trigonometric Identities are useful whenever trigonometric functions are involved in an expression or an equation. Trigonometric Identities are true for every value of variables occurring on both sides of an equation. Geometrically, these identities involve certain trigonometric functions (such as sine, cosine, tangent) of one or more angles.

Sine, cosine and tangent are the primary trigonometry functions whereas cotangent, secant and cosecant are the other three functions. The trigonometric identities are based on all the six trig functions. Check Trigonometry Formulas to get formulas related to trigonometry.

What are Trigonometric Identities?

Trigonometric Identities are the equalities that involve trigonometry functions and holds true for all the values of variables given in the equation.

There are various distinct trigonometric identities involving the side length as well as the angle of a triangle. The trigonometric identities hold true only for the right-angle triangle.

All the trigonometric identities are based on the six trigonometric ratios. They are sine, cosine, tangent, cosecant, secant, and cotangent. All these trigonometric ratios are defined using the sides of the right triangle, such as an adjacent side, opposite side, and hypotenuse side. All the fundamental trigonometric identities are derived from the six trigonometric ratios.

Trigonometric Identities PDF

Click here to download the PDF of trigonometry identities of all functions such as sin, cos, tan and so on.

List of Trigonometric Identities

There are various identities in trigonometry which are used to solve many trigonometric problems. Using these trigonometric identities or formulas, complex trigonometric questions can be solved quickly. Let us see all the fundamental trigonometric identities here.

Reciprocal Trigonometric Identities

The reciprocal trigonometric identities are:

• Sinθ = 1/Cscθ or Cscθ = 1/Sinθ
• Cosθ = 1/Secθ or Secθ = 1/Cosθ
• Tanθ = 1/Cotθ or Cotθ = 1/Tanθ

Pythagorean TrigonometricIdentities

There are three Pythagorean trigonometric identities in trigonometry that are based on the right-triangle theorem or Pythagoras theorem.

• sin²a + cos²a = 1
• 1+tan² a = sec² a
• cosec²a = 1 + cot²a

RatioTrigonometric Identities

The trigonometric ratio identities are:

• Tanθ = Sinθ/Cosθ
• Cotθ = Cosθ/Sinθ

Trigonometric Identities of Opposite Angles

The list of opposite angle trigonometric identities are:

• Sin (-θ) = – Sinθ
• Cos (-θ) = Cosθ
• Tan (-θ) = – Tanθ
• Cot (-θ) = – Cotθ
• Sec (-θ) = Secθ
• Csc (-θ) = -Cscθ

Trigonometric Identities of Complementary Angles

In geometry, two angles are complementary if their sum is equal to 90 degrees. Similarly, when we can learn here the trigonometric identities for complementary angles.

• Sin (90 –θ) = Cosθ
• Cos (90 –θ) = Sinθ
• Tan (90 –θ) = Cotθ
• Cot ( 90 –θ) = Tanθ
• Sec (90 –θ) = Cscθ
• Csc (90 –θ) = Secθ

Trigonometric Identities of Supplementary Angles

Two angles are supplementary if their sum is equal to 90 degrees. Similarly, when we can learn here the trigonometric identities for supplementary angles.

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

Sum and Difference of Angles Trigonometric Identities

Consider two angles ,α and β, the trigonometric sum and difference identities are as follows:

• sin(α+β)=sin(α).cos(β)+cos(α).sin(β)
• sin(α–β)=sinα.cosβ–cosα.sinβ
• cos(α+β)=cosα.cosβ–sinα.sinβ
• cos(α–β)=cosα.cosβ+sinα.sinβ

• \(\begin{array}{l}\tan (\alpha + \beta) = \frac{\tan \alpha + \tan \beta}{1 – \tan \alpha. \tan \beta}\end{array} \)

• \(\begin{array}{l}\tan (\alpha – \beta) = \frac{\tan \alpha – \tan \beta}{1 + \tan \alpha. \tan \beta}\end{array} \)

Double Angle Trigonometric Identities

If the angles are doubled, then the trigonometric identities for sin, cos and tan are:

• sin 2θ = 2 sinθ cosθ
• cos 2θ = cos²θ – sin²θ = 2 cos²θ – 1 = 1 – 2sin² θ
• tan 2θ = (2tanθ)/(1 – tan²θ)

Half Angle Identities

If the angles are halved, then the trigonometric identities for sin, cos and tan are:

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

Product-Sum Trigonometric Identities

The product-sum trigonometric identities change the sum or difference of sines or cosines into a product of sines and cosines.

• Sin A + Sin B = 2 Sin(A+B)/2 . Cos(A-B)/2
• Cos A + Cos B = 2 Cos(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 Sin(A+B)/2 . Sin(A-B)/2

Trigonometric Identities of Products

These identities are:

• Sin A. Sin B = [Cos (A – B) – Cos (A + B)]/2
• Sin A. Cos B = [Sin (A + B) + Sin (A – B)]/2
• Cos A. Cos B = [Cos (A + B) + Cos (A – B)]/2

TrigonometricIdentities Proofs

Similarly, an equation that involves trigonometric ratios of an angle represents a trigonometric identity.
The upcoming discussion covers the fundamental trigonometric identities and their proofs.
Consider the right angle ∆ABCwhich is right-angled at B as shown in the given figure.

Applying Pythagoras Theorem for the given triangle, we have

(hypotenuse)² = (base)² + (perpendicular)²

AC² = AB²+BC² ………………………..(1)

Let us prove the three Pythagoras trigonometric identities, which are commonly used.

Trigonometric Identity 1

Now, divide each term of equation (1) by AC², we have

\(\begin{array}{l}\frac{AC^2}{AC^2} = \frac{AB^2}{AC^2}~+~\frac{BC^2}{AC^2}\end{array} \)

\(\begin{array}{l}\Rightarrow \frac{AB^2}{AC^2}+\frac{BC^2}{AC^2} = 1\end{array} \)

\(\begin{array}{l}\Rightarrow (\frac{AB}{AC})^2+(\frac{BC}{AC})^2 = 1 …(2)\end{array} \)

We know that,

\(\begin{array}{l}(\frac{AB}{AC})^2 = \cos a\ and\ (\frac{BC}{AC})^2 = \sin a \end{array} \)

Thus equation (2) can be written as-

sin²a + cos²a = 1

Identity 1 is valid for angles 0 ≤ a ≤ 90.

Trigonometric Identity 2

Now Dividing the equation (1) by AB², we get

\(\begin{array}{l}\frac{AC^2}{AB^2}= \frac{AB^2}{AB^2}+\frac{BC^2}{AB^2}\end{array} \)

\(\begin{array}{l}\Rightarrow \frac{AC^2}{AB^2}= 1+\frac{BC^2}{AB^2}\end{array} \)

\(\begin{array}{l}\Rightarrow (\frac{AC}{AB})^2 = 1+(\frac{BC}{AB})^2 …(3)\end{array} \)

By referring to trigonometric ratios, it can be seen that:

\(\begin{array}{l}\frac{AC}{AB} = \frac{hypotenuse}{side~ adjacent~ to ~angle~ a} = \sec a\end{array} \)

Similarly,

\(\begin{array}{l}\frac{BC}{AB} = \frac{side~ opposite~ to~ angle~ a}{side ~adjacent~ to~ angle~ a} = \tan a\end{array} \)

Replacing the values of AC/AB and BC/AB in equation (3) gives,

1+tan² a = sec² a

As it is known that tan a is not defined for a = 90°, therefore, identity 2 obtained above is true for 0 ≤ A <90.

Trigonometric Identity 3

Dividing the equation (1) by BC², we get

\(\begin{array}{l}\frac{AC^2}{BC^2} = \frac{AB^2}{BC^2}~ +~\frac{BC^2}{BC^2}\end{array} \)

\(\begin{array}{l}\Rightarrow \frac{AC^2}{BC^2} = \frac{AB^2}{BC^2}+1\end{array} \)

\(\begin{array}{l}\Rightarrow (\frac{AC}{BC})^2 = (\frac{AB}{BC})^2+1 …(4)\end{array} \)

By referring to trigonometric ratios, it can be seen that:

\(\begin{array}{l}\frac{AC}{BC} = \frac{hypotenuse}{side ~opposite~ to~ angle~ a} = cosec\ a\end{array} \)

Also,

\(\begin{array}{l}\frac{AB}{BC} = \frac{side~ adjacent~ to~ angle~ a}{side~ opposite ~to ~angle~ a} = \cot a\end{array} \)

Replacing the values of AC/BC and AB/BCin the equation (4) gives,

cosec²a = 1 + cot²a

Since cosec a and cot a are not defined for a = 0°. Therefore the identity 3 is obtained is true for all the values of ‘a’ except at a = 0°. Therefore, the identity is truefor all such that, 0° < a ≤ 90°.

Triangle Identities (Sine, Cosine, Tangent rule)

If the identities or equations are applicable for all the triangles and not just for right triangles, then they are the triangle identities. These identities will include:

• Sine law
• Cosine law
• Tangent law

If A, B and C are the vertices of a triangle and a, b and c are the respective sides, then;

According to the sine law or sine rule,

\(\begin{array}{l}\frac{a}{Sin A} = \frac{b}{Sin B} = \frac{c}{Sin C}\end{array} \)

Or

\(\begin{array}{l}\frac{Sin A}{a} = \frac{Sin B}{b} = \frac{Sin C}{c}\end{array} \)

According to cosine law,

\(\begin{array}{l}c^{2}=a^{2}+b^{2}-2 a b \cos C\end{array} \)

Or

\(\begin{array}{l}cos~ C=\frac{a^{2}+b^{2}-c^{2}}{2 a b}\end{array} \)

According to tangent law,

\(\begin{array}{l}\frac{a-b}{a+b}=\frac{\tan \left(\frac{A-B}{2}\right)}{\tan \left(\frac{A+B}{2}\right)}\end{array} \)

Video Lessons on Trigonometric Identities