Question

sinxtan(x/2)+2cosx=2/1+tan^2(x/2)​

Answer 1

$\large{\underline{\boxed{\purple{ANSWER}}}}$

TO PROVE:-

$\sin(x) \tan( \frac{x}{2} ) + 2 \cos(x ) = \frac{2}{1 + \tan^{2} ( \frac{x}{2} ) }$

IDENTITIES TO BE USED:-

$\sin(2x) = \frac{2 \tan(x) }{1 + \tan ^{2} (x) }$

AND,

$\cos(2x) = \frac{1 - \tan ^{2} (x) }{1 + \tan ^{2} (x) }$

$\large{\underline{\boxed{\red{SOLUTION}}}}$

ON L.H.S WE HAVE ,

$\sin( {x} ) \tan( \frac{x}{2} ) + 2 \cos(x)$

So, using above identities , we get

$\tan( \frac{x}{2} ) ( \frac{2 \tan( \frac{x}{2} ) }{1 + \tan ^{2} ( \frac{x}{2} ) } ) + 2( \frac{1 - \tan ^{2} ( \frac{x}{2} ) }{1 + \tan ^{2} ( \frac{x}{2} ) } ) \\ \\ \\ \\ = > \frac{2 \tan ^{2} ( \frac{x}{2} ) - 2 \tan ^{2} ( \frac{x}{2} ) + 2} {1 + \tan ^{2} ( \frac{x}{2} ) } \\ \\ \\ \\ = > \frac{2}{1 + \tan ^{2} ( \frac{x}{2} ) }$

ALSO ON R.H.S , WE HAVE

$\frac{2}{1 + \tan ^{2} ( \frac{x}{2} ) }$

HENCE ,

LHS = RHS

HENCE PROVED.