Trigonometric Identities

Trigonometric identities come into play whenever trigonometric functions appear in an expression or equation. They hold true for every possible value of the variables involved on both sides of the equation. Geometrically, these identities relate to specific trigonometric functions like sine, cosine, and tangent, which involve one or more angles.

The primary trigonometric functions are sine, cosine, and tangent, while cotangent, secant, and cosecant are the complementary functions. These identities are established based on all six trigonometric functions.

In this article, we will cover the concept of trigonometric identities. This category falls under the broader category of Trigonometry, which is a crucial Chapter in class 11 Mathematics. It is not only essential for board exams but also for competitive exams like the Joint Entrance Examination(JEE Main) and other entrance exams such as SRMJEE, BITSAT, WBJEE, BCECE, and more. Questions based on this topic have been asked frequently in JEE Mains.

What are Trigonometric Identities?

Trigonometric identities are equations involving trigonometric functions that remain true for all values of the variables in the equation. These identities involve various relationships between the sides and angles of a triangle, particularly in the context of right-angle triangles. They rely on the fundamental trigonometric ratios: sine, cosine, tangent, cosecant, secant, and cotangent. These ratios are defined based on the sides of a right triangle — the adjacent side, opposite side, and hypotenuse.

Fundamental trigonometric identities are derived directly from these ratios, forming the basis for solving trigonometric problems across mathematics and science.

List of Trigonometric Identities

In trigonometry, there are numerous identities that help in solving a variety of problems. These formulas are essential for tackling complex trigonometric questions efficiently. They provide fundamental tools that enable quick solutions to various trigonometric problems. Let's explore the essential trigonometric identities used in these calculations.

1. Pythagorean theorem identities

The trigonometric identities that follow Pythagoras's theorem are called Pythagorean theorem identities. There are three trigonometric identities that are based on the Pythagoras theorem.

$\begin{aligned} & \sin ^2 t+\cos ^2 t=1 \\ & 1+\tan ^2 t=\sec ^2 t \\ & 1+\cot ^2 t=\csc ^2 t \\ & \tan t=\frac{\sin t}{\cos t}, \quad \cot t=\frac{\cos t}{\sin t}\end{aligned}$

2. Reciprocal identities

Trigonometric ratios which are reciprocal to other trigonometric ratios are called reciprocal identities.

The reciprocal trigonometric identities are

$\begin{aligned} & \text { Cosecant } \quad \csc t=\frac{\text { hypotenuse }}{\text { opposite }}=\frac{1}{\sin t} \\ & \text { Secent } \quad \sec t=\frac{\text { hypotenuse }}{\text { adjacent }}=\frac{1}{\cos t} \\ & \text { Cotangent } \quad \cot t=\frac{\text { adjacent }}{\text { opposite }}=\frac{1}{\tan t}\end{aligned}$

3. Addition of angles trignometric identites

There are three trigonometric identities related to the Addition of angles.
If $A$ and $B$ are two angles.
a) $\sin (A+B)=\sin A \cos B+\cos A \sin B$
b) $C o s(A+B)=\cos A \cos B-\sin A \sin B$

c) $\tan (A+B)=\frac{\tan A+\tan B}{1-\tan A \tan B}$

4. Difference of angles trigonometric identities

|There are three trigonometric identities related to the difference in angles.
If $A$ and $B$ are two angles.
a) $\sin (A-B)=\sin A \cos B-C \cos A \sin B$
b) $\operatorname{Cos}(A-B)=\cos A \cos B-\operatorname{Sin} A \sin B$

c) $\tan (A-B)=\frac{\tan A-\tan B}{1+\tan A \tan B}$

5. Double angle trignometric identities

The double-angle trignometric identities is used to transform the trignometric ratios of double angles into the trignometric ratios of single angles.
If A is the angle of the triangle,
a) $\sin 2 A=2 \sin A \cos A$

b) $\cos 2 \mathrm{~A}=\cos ^2 A-\sin ^2 A=2 \cos ^2 A-1=1-2 \sin ^2 A$

c) $\tan 2 A=\frac{2 \tan A}{1-\tan ^2 A}$

6. Trigonometric identities of allied angles

Two angles are called allied if their sum or difference is a multiple of $\pi / 2(90)$
Below are some trigonometric identities of allied angles
- $\sin (90-\theta)=\cos (\theta)$
- $\cos (90-\theta)=\sin (\theta)$
- $\tan (90-\theta)=\cot (\theta)$
- $\csc (90-\theta)=\sec (\theta)$
- $\sec (90-\theta)=\csc (\theta)$
- $\cot (90-\theta)=\tan (\theta)$
- $\sin (90+\theta)=\cos (\theta)$
- $\cos (90+\theta)=-\sin (\theta)$
- $\tan (90+\theta)=-\cot (\theta)$
- $\csc (90+\theta)=\sec (\theta)$
- $\sec (90+\theta)=-\csc (\theta)$
- $\cot (90+\theta)=-\tan (\theta)$

7. Triple-angle Trigonometric identities

If angles are triple then trigonometric identities of $\sin \mathrm{A}, \cos \mathrm{A}$, and $\tan \mathrm{A}$ are
If A is the angle of the triangle,

$\begin{aligned} & \sin 3 \mathrm{~A}=3 \sin \mathrm{A}-4 \sin ^3 \mathrm{~A} \\ & \cos 3 A=4 \cos ^3 A-3 \cos A \\ & \tan 3 A=\frac{3 \tan A-\tan ^3 A}{1-3 \tan ^2 A} \\ & \cot 3 \mathrm{~A}=\frac{\cot ^3 \mathrm{~A}-3 \cot \mathrm{A}}{3 \cot ^2 \mathrm{~A}-1}\end{aligned}$

8. Trigonometric identities of supplementary angles

Two angles are called supplementary angles if their sum is 180.
Below are the trigonometric identities of supplementary angles:
- $\sin (180-\theta)=\sin (\theta)$
- $\cos (180-\theta)=-\cos (\theta)$
- $\tan (180-\theta)=-\tan (\theta)$
- $\csc (180-\theta)=\csc (\theta)$
- $\sec (180-\theta)=-\sec (\theta)$
- $\cot (180-\theta)=-\cot (\theta)$

9. Sum-product trigonometric identities

The Sum/Difference identity is used to express the sum or difference of sine and cosine functions into the Product of sine and cosine functions. Below are some Product sum trigonometric identities :

$\begin{aligned} & \sin \mathrm{C}+\sin \mathrm{D}=2 \sin \frac{\mathrm{C}+\mathrm{D}}{2} \cos \frac{\mathrm{C}-\mathrm{D}}{2} \\ & \sin \mathrm{C}-\sin \mathrm{D}=2 \cos \frac{\mathrm{C}+\mathrm{D}}{2} \sin \frac{\mathrm{C}-\mathrm{D}}{2} \\ & \cos \mathrm{C}+\cos \mathrm{D}=2 \cos \frac{\mathrm{C}+\mathrm{D}}{2} \cos \frac{\mathrm{C}-\mathrm{D}}{2} \\ & \cos \mathrm{C}-\cos \mathrm{D}=-2 \sin \frac{\mathrm{C}+\mathrm{D}}{2} \sin \frac{\mathrm{C}-\mathrm{D}}{2}\end{aligned}$

10. Trigonometric identities of the opposite angle

Below are some trigonometric identities of opposite angles:
$\begin{aligned}
& \operatorname{Sin}(-A)=-\sin A \\
& \operatorname{Cos}(-A)=\cos A \\
& \operatorname{Tan}(-A)=-\operatorname{Tan} A \\
& \operatorname{Cot}(-A)=-\cot A \\
& \operatorname{Sec}(-A)=\sec A\\
&\operatorname{Cosec}(-A)=-\operatorname{cosec} A
\end{aligned}$
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:
1) Sine Law
2) Cosine Law
3) Tangent Law

Sine Rule
The ratio of the sine of one of the angles to the length of its opposite side will be equal to the other two ratios of the sine of the angle measured to the opposite side.

$
\frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c}
$

Cosine Rule
For a triangle with angles $\mathrm{A}, \mathrm{B}$, and C , and opposite corresponding sides $\mathrm{a}, \mathrm{b}$, and c , respectively, the Law of Cosines is given as three equations.

$
\cos A=\frac{b^2+c^2-a^2}{2 b c} \quad \cos B=\frac{a^2+c^2-b^2}{2 a c} \quad \cos C=\frac{a^2+b^2-c^2}{2 a b}
$

Tangent Rule or Napier's Analogy
For any $\triangle A B C$,

$
\tan \left(\frac{A-B}{2}\right)=\frac{a-b}{a+b} \cot \frac{C}{2}
$

Solved Examples Based on Trigonometric Identities

Example 1: For a triangle $A B C$, the value of $\cos 2 A+\cos 2 B+\cos 2 C$ is least. If its inradius is 3 and incentre is $M$, then which of the following is NOT correct? [JEE MAINS 2023]
1) The perimeter of $\triangle A B C$ is $18 \sqrt{3}$
2) $\sin 2 A+\sin 2 B+\sin 2 C=\sin A+\sin B+\sin C$
3) $\overrightarrow{M A} \cdot \overrightarrow{M B}=-18$
4) The area of $\triangle A B C$ is $\frac{27 \sqrt{3}}{2}$

Solution:
Let P
$=\cos 2 \mathrm{~A}+\cos 2 \mathrm{~B}+\cos 2 \mathrm{C}$
$=2 \cos (\mathrm{A}+\mathrm{B}) \cos (\mathrm{A}-\mathrm{B})+2 \cos ^2 \mathrm{C}-1$
$=2 \cos (\pi-\mathrm{C}) \cos (\mathrm{A}-\mathrm{B})+2 \cos ^2 \mathrm{C}-1$
$=-2 \cos \mathrm{C}[\cos (\mathrm{A}-\mathrm{B})+\cos (\mathrm{A}+\mathrm{B})]-1$
$=-1-4 \cos \mathrm{A} \cos \mathrm{B} \cos \mathrm{C}$
for $P$ to be the minimum
$\cos \mathrm{A} \cos \mathrm{B} \cos \mathrm{C}$ must be maximum
$\Rightarrow \triangle \mathrm{ABC}$ is an equilateral triangle

Example 3: If $15 \sin ^4 \alpha+10 \cos ^4 \alpha=6$ for some $\alpha \epsilon R$, then the value of $27 \sec ^6 \alpha+8 \operatorname{cosec}^6 \alpha$ is equal to: [JEE MAINS 2021]
1) 350
2) 250
3) 400
4) 500

Solution
$
\begin{aligned}
& 15 \sin ^4 \alpha+10 \cos ^4 \alpha=6 \\
& 15 \sin ^4 \alpha+10 \cos ^4 \alpha=6\left(\sin ^2 \alpha+\cos ^2 \alpha\right)^2 \\
& \left(3 \sin ^2 \alpha-2 \cos ^2 \alpha\right)^2=0 \\
& \tan ^2 \alpha=\frac{2}{3} \cdot \cot ^2 \alpha=\frac{3}{2} \\
& \Rightarrow 27 \sec ^6 \alpha+8 \operatorname{cosec}^6 \alpha \\
& =27\left(\sec ^6 \alpha\right)^3+8\left(\operatorname{cosec}^6 \alpha\right)^3 \\
& =27\left(1+\tan ^2 \alpha\right) 3+8\left(1+\cot ^2 \alpha\right)^3 \\
& =250
\end{aligned}
$

Hence, the answer is option 3.

Example 4: Let $S=\left\{\theta \epsilon[-2 \pi, 2 \pi]: 2 \cos ^2 \theta+3 \sin \theta=0\right\}$. Then the sum of the elements of $S$ is: [JEE MAINS 2019]
1) $\frac{13 \pi}{6}$
2) $\frac{5 \pi}{3}$
3) $2 \pi$
4) $\pi$

Solution:

Given equation

$
\begin{aligned}
& 2 \cos ^2 \theta+3 \sin \theta=0 \\
& 2-2 \sin ^2 \theta+3 \sin \theta=0
\end{aligned}
$

Or
$
\begin{aligned}
& 2 \sin ^2 \theta-3 \sin \theta-2=0 \\
& \sin \theta=-\frac{1}{2} \text { or } 2 \\
& \sin \theta \neq 2
\end{aligned}
$
because solution is $[-2 \pi, 2 \pi]$
$
\begin{aligned}
& -\pi+\frac{\pi}{6}, \frac{-\pi}{6}, \pi+\frac{\pi}{6}, 2 \pi-\frac{\pi}{6} \\
& \text { Sum }=2 \pi
\end{aligned}
$

Hence, the answer is option 3.

Example 5: If $\frac{\sqrt{2} \sin \alpha}{\sqrt{1+\cos 2 \alpha}}=\frac{1}{7}$ and $\sqrt{\frac{1-\cos 2 \beta}{2}}=\frac{1}{\sqrt{10}}, \alpha, \beta \space\epsilon\left(0, \frac{\pi}{2}\right)$, then $\tan (\alpha+2 \beta)$ is equal to

1) 0

2) 1

3) 0.5

4) 2

Solution:

$\begin{aligned} & \frac{\sqrt{2} \sin \alpha}{\sqrt{2} \cos \alpha}=\frac{1}{7} \\ & \tan \alpha=\frac{1}{7} \\ & \sin \beta=\frac{1}{\sqrt{10}} \\ & \tan \beta=\frac{1}{\sqrt{3}} \\ & \tan 2 \beta=\frac{2 \cdot \frac{1}{3}}{1-\frac{1}{9}}=\frac{\frac{2}{3}}{\frac{8}{9}}=\frac{3}{4} \\ & \tan (\alpha+2 \beta)=\frac{\tan \alpha+\tan 2 \beta}{1-\tan \alpha \tan 2 \beta}=1\end{aligned}$

Hence, the answer is option 2.

Summary

Trigonometric identities are crucial tools that help us understand and use trigonometry effectively. They simplify complex calculations, making it easier to apply trigonometry in mathematics and science. These identities play a key role in advancing our knowledge and capabilities in these fields.