Question

The value of $\frac{\sin 5x-2\sin 3x+\sin x}{\cos 5x-\cos x}$ is-

• A. $\tan 5x$
• B. $\tan x$
• C. $\sin x$
• D. None of these

Answer 1

The correct option is B $\tan x$

The value of $\frac{2\sin 5x-2\sin 3x+\sin x}{cos5x-cosx}$
$=\frac{2\sin \left(\frac{5x-x}{2}\right)cos\left(\frac{5x-x}{2}\right)-2\sin 3x}{2\sin \left(\frac{5x-x}{2}\right)\sin \left(\frac{x-5x}{2}\right)}$
$=\frac{2\sin 3xcos2x-2\sin 3x}{-2\sin 3x\sin 2x}=\frac{cos2x-1}{-2\sin 2x}$
$=\frac{1-2{\sin }^{2}x-1}{-2\sin xcosx}=\tan x$