Question

prove that -cot x = sin 3x + sin x / cos 3x - cos x​

Answer 1

$\huge \sf \red {\underline { Appreciate \: Question}} </p><p>$

To Prove That :-

$\\( sin3x \: + sinx)sin(x) + (cos3x - cosx)cosx \: = 0$

$\huge \sf \red {\underline {Solution \div }} </p><p>$

Using identity :-

$\ \sf \: \sin(x) + \sin(y) = \\ 2sin( \frac{x + y}{2} )cos( \frac{x - y}{2} ) \\ \large \ \sf \: or \\ \cos(x) + \cos(y) = 2sin( \frac{x + y}{2} )sin( \frac{x - y}{2} )$

L.H.S. =

$\\ \\ = [2sin( \frac{3x + x}{2} )cos( \frac{3x - x}{2} )] - \\ \\ = [2sin( \frac{3x + x}{2} )sin( \frac{3x - x}{2} )] \\ \\ = sin2xcos2xsinx - 2sin2xcosxsinx$

= 0 = R.H.S

LHS = RHS

_________________________________