# Trig Identities

Trigonometry is one of the main topics in math, and many students have issues with trigonometric topics. One of the main topics is trig identities. These trigonometric identities help us with many other topics, including trigonometric equations, derivative with trigonometric functions, integral of trigonometric functions and proving trig identities.

Here we have the list of main trig identities:

Reciprocal identities:

• $\sin u=\frac{1}{\csc u}$
• $\cos u=\frac{1}{\sec u}$
• $\tan u=\frac{1}{\cot u}$
• $\cot u=\frac{1}{\tan u}$
• $\csc u=\frac{1}{\sin u}$
• $\sec u=\frac{1}{\cos u}$

Pythagorean Identities:

• $\sin^{2} u+\cos^{2} u=1$
• $1+\tan^{2} u=\sec^{2} u$
• $1+\cot^{2} u=\csc^{2} u$

Quotient Identities:

• $\tan u=\frac{\sin u}{\cos u}$
• $\cot u=\frac{\cos u}{\sin u}$

Co-Function Identities:

• $\sin \left(\frac{\pi}{2}-u\right)=\cos u$
• $\cos \left(\frac{\pi}{2}-u\right)=\sin u$
• $\tan \left(\frac{\pi}{2}-u\right)=\cot u$
• $\cot \left(\frac{\pi}{2}-u\right)=\tan u$
• $\csc \left(\frac{\pi}{2}-u\right)=\sec u$
• $\sec \left(\frac{\pi}{2}-u\right)=\csc u$

Parity Identities: We know that $\sin x$ is an odd function and $\cos x$ is an even function.

• $\sin (-u)=-\sin u$
• $\cos (-u)=\cos u$
• $\tan (-u)=-\tan u$
• $\cot (-u)=-\cot u$
• $\csc (-u)=-\csc u$
• $\sec (-u)=\sec u$

Sum and Difference Formulas:

• $\sin (u \pm v)=\sin u \cos v \pm \cos u \sin v$
• $\cos (u \pm v)=\cos u \cos v \mp \sin u \sin v$
• $\tan (u \pm v)=\frac{\tan u \pm \tan v}{1 \mp \tan u \tan v}$

Double Angle Formulas:

• $\sin (2 u) =2 \sin u \cos u$
• $\cos (2 u) =\cos ^{2} u-\sin ^{2} u$
• $\cos (2 u) =2 \cos ^{2} u-1$
• $\cos (2 u) =1-2 \sin ^{2} u$
• $\tan (2 u) =\frac{2 \tan u}{1-\tan ^{2} u}$

To see more examples of Double Angle Formulas visit this page.

Half Angle Formulas:

• $\sin^{2} u=\frac{1-\cos (2 u)}{2}$
• $\cos^{2} u=\frac{1+\cos (2 u)}{2}$
• $\tan^{2} u=\frac{1-\cos (2 u)}{1+\cos (2 u)}$

Sum to product formulas:

• $\sin u+\sin v=2 \sin \left(\frac{u+v}{2}\right) \cos \left(\frac{u-v}{2}\right)$
• $\sin u-\sin v=2 \cos \left(\frac{u+v}{2}\right) \sin \left(\frac{u-v}{2}\right)$
• $\cos u+\cos v=2 \cos \left(\frac{u+v}{2}\right) \cos \left(\frac{u-v}{2}\right)$
• $\cos u-\cos v=-2 \sin \left(\frac{u+v}{2}\right) \sin \left(\frac{u-v}{2}\right)$

Product to Sum Formulas:

• $\sin u \sin v=\frac{1}{2}[\cos (u-v)-\cos (u+v)]$
• $\cos u \cos v=\frac{1}{2}[\cos (u-v)+\cos (u+v)]$
• $\sin u \cos v=\frac{1}{2}[\sin (u+v)+\sin (u-v)]$
• $\cos u \sin v=\frac{1}{2}[\sin (u+v)-\sin (u-v)]$

If you need help to understand the trig identities, contact us.

## 2 thoughts on “Trig Identities”

1. […] to use identities. One of the main and crucial categories of identities is Double Angle Formula. In this post, we talked about different trig identities and formulas. Double Angle Formulas are trigonometric […]

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2. […] and one of the most important identities in trigonometric functions is the half angle formula. In this post, we talked about different trig identities and formulas. First we recommend to visit the previous […]

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