Question

help me please in proving ​

Answer 1

LHS:-

$\cot \theta - \tan \theta$

$= \frac{ \cos \theta}{ \sin \theta} - \frac{ \sin \theta}{ \cos \theta }$

$= \frac{ \cos^{2} - { \sin }^{2} \theta }{ \sin \theta \cos \theta }$

Now,

$\sin2 \theta = 2 \sin\theta \cos \theta$

$\implies \sin \theta \cos \theta = \frac{2 \sin \theta }{2}$

Also,

$\cos2\theta = \cos^{2} \theta - \sin^{2} \theta$

Putting all the values, we get,

$\frac{ \cos 2\theta}{ \frac{ \sin2 \theta }{2} }$

$= \frac{2 \cos2 \theta }{ \sin2 \theta }$

$= 2 \cot \theta$

RHS:-

$= 2 \cot2 \theta$

Hence, LHS=RHS (Proved).