Question

prove that
$\tan(x) = \frac{ \sin(2x) }{2 \cos ^{2} (x) }$

Answer 1

$l.h.s. = \: \tan(x) = \frac{ \sin(x) }{ \cos(x) } \\ \\ = \frac{2 \sin(x) \cos(x) }{2 \cos(x) \cos(x) } \\ \\ but \: \: 2 \sin(x) \cos(x) = \sin(2x) \\ \\ therefore \\ \\ \frac{2 \sin(x) \cos(x) }{2 \cos(x) \cos(x) } = \frac{ \sin(2x) }{2 \cos {}^{2} (x) } = r.h.s. \\ \\ proved \: \\ \\ hope \: it \: helps...$