# 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:

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

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:

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