Question

prove that sin2 theta/1+cos 2theta is tan theta​

Answer 1

Correct Question:-

➡ Prove that, $\sf \frac{sin(2x)}{1+cos(2x)} = tan(x)$

Proof:-

Taking LHS,

$\sf \frac{ \sin(2x) }{1 + \cos(2x) }$

$\sf = \frac{ \sin(2x) }{1 + \cos(2x) } \times \frac{1 - \cos(2x) }{1 - \cos(2x) }$

$\sf = \frac{ \sin(2x)(1 - \cos(2x)) }{ {(1)}^{2} - { \cos}^{2}(2x) }$

$\sf = \frac{ \sin(2x)(1 - \cos(2x)) }{ \sin^{2}(2x) }$

$\sf = \frac{ \cancel{\sin(2x)}(1 - \cos(2x)) }{ \cancel{\sin(2x) }\times \sin(2x) }$

$\sf = \frac{1 - \cos(2x) }{ \sin(2x) }$

$\sf = \frac{1 - (1 - { 2\sin }^{2}(x))}{2 \sin(x) \cos(x) }$

$\sf = \frac{ \cancel1 - \cancel1 + 2{ \sin }^{2}(x)}{2 \sin(x) \cos(x) }$

$\sf = \frac{ \cancel2{ \sin }^{2}(x)}{ \cancel2 \sin(x) \cos(x) }$

$\sf = \frac{ \sin(x) }{ \cos(x) }$

$\sf = \tan(x)$

Taking RHS,

$\sf = \tan(x)$

Hence,

LHS = RHS (Hence Proved)

Formulae Used:-

$\sf \mapsto { \sin}^{2}(x) + \cos^{2} (x) = 1 \implies{ \sin}^{2}(x) = 1 - \cos^{2} (x)$

$\sf \mapsto \cos(2x) = 1 - 2 \sin ^{2} (x)$

$\sf \mapsto \sin(2x) = 2 \sin(x) \cos(x)$