Question

Prove the identity.

$\cos \left(x+\frac{\pi }{3}\right)+\sin \left(x-\frac{\pi }{6}\right)=0$

Short answer

$\cos \left(x+\frac{\pi }{3}\right)+\sin \left(x-\frac{\pi }{6}\right)=0$

Step-by-step solution

Step 1. Given information.

The given equation, $\cos \left(x+\frac{\pi }{3}\right)+\sin \left(x-\frac{\pi }{6}\right)=0$.

Addition and Subtraction formulas used are:

$\begin{array}{l}\cos (A+B)=\cos A\cos B-\sin A\sin B\\ \sin (A-B)=\sin A\cos B-\cos A\sin B\end{array}$

Step 2. Evaluate to get the final answer.

The identity is proved as:

$\begin{array}{l}LHS\\ =\cos \left\{x+\frac{\pi }{3}\right\}+\sin \left\{x-\frac{\pi }{6}\right\}\\ =\left\{\cos x\times \cos \frac{\pi }{3}-\sin x\times \sin \frac{\pi }{3}\right\}+\left\{\sin x\times \cos \frac{\pi }{6}-\cos x\times \sin \frac{\pi }{6}\right\}\\ =\left\{\cos x\times \frac{1}{2}-\sin x\times \frac{\sqrt{3}}{2}\right\}+\left\{\sin x\times \frac{\sqrt{3}}{2}-\cos x\times \frac{1}{2}\right\}\\ =0\\ =RHS\end{array}$