# How do you verify sin(x + pi/6) - cos(x + pi/3) = sqrt3 sin x?

Apr 8, 2018

It is given that the LHS is $\sin \left(x + \frac{\pi}{6}\right) - \cos \left(x + \frac{\pi}{3}\right)$

Applying, color(magenta)(sin(A+B) = sinAcosB + cosAsinB and color(red)(cos(A+B) = cosAcosB - sinAsinB

color(white)(dd

$\implies \textcolor{m a \ge n t a}{\sin x \cos \left(\frac{\pi}{6}\right) + \cos x \sin \left(\frac{\pi}{6}\right)} - \left(\textcolor{red}{\cos x \cos \left(\frac{\pi}{3}\right) - \sin x \sin \left(\frac{\pi}{3}\right)}\right)$

color(white)(dd

$\implies \sin x \left(\frac{\sqrt{3}}{2}\right) + \cos x \left(\frac{1}{2}\right) - \cos x \left(\frac{1}{2}\right) + \sin x \left(\frac{\sqrt{3}}{2}\right)$

color(white)(dd

$\implies \cancel{2} \times \sin x \left(\frac{\sqrt{3}}{\cancel{2}}\right)$

$\implies \sqrt{3} \sin x$

Apr 8, 2018

See below.

#### Explanation:

Identities:

$\textcolor{red}{\boldsymbol{\sin \left(A + B\right) = \sin A \cos B + \cos A \sin B}}$

$\textcolor{red}{\boldsymbol{\cos \left(A + B\right) = \cos A \cos B - \sin A \sin B}}$

$L H S :$

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

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

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

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

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

$\cos \left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2}$
$\cos \left(\frac{\pi}{3}\right) = \frac{1}{2}$
$\sin \left(\frac{\pi}{6}\right) = \frac{1}{2}$
$\sin \left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2}$

$\sin \left(x\right) \frac{\sqrt{3}}{2} + \cos \left(x\right) \left(\frac{1}{2}\right) - \cos \left(x\right) \left(\frac{1}{2}\right) + \sin \left(x\right) \frac{\sqrt{3}}{2}$

$2 \sin \left(x\right) \frac{\sqrt{3}}{2}$

$\sqrt{3} \sin \left(x\right)$

$L H S \equiv R H S$