Question

prove that
$1 + \sin(2x) = ( \sin(x) + \cos(x) )^{2}$

Answer 1

$since \: \sin(2x) = 2 \sin(x) \cos(x) \\ \\ and \: \sin ^{2} (x) + \cos {}^{2} (x) = 1 \\ \\ therefore$
L. H. S.
$= 1 + \sin(2x) = \sin {}^{2} (x) + \cos {}^{2} (x) + 2 \sin(x) \cos(x) \\ \\ using \: identity \: a {}^{2} + b {}^{2} + 2ab = (a + b) {}^{2} \\ \\ therefore \\ \\ \sin {}^{2} (x) + \cos {}^{2} (x) + 2 \sin(x) \cos(x) \\ \\ = ( \sin(x) + \cos(x) ) {}^{2} = r.h.s. \\ \\ proved \\ \\ \\ hope \: it \: helps$