Question

Pls solve the first question of tis page

Answer 1

$\sin(x) + \sin(2x) + \sin(4x) + \sin(5x) = 4 \cos( \frac{x}{2} ) \cos( \frac{3x}{2} ) \sin(3x) \\ lhs = \\ \sin(x) + \sin(2x) + \sin(4x) + \sin(5x) \\ = ( \sin(5x) + \sin(x) ) + ( \sin(2x) + \sin(4x) ) \\ = 2 \sin(3x) \cos(2x) + 2 \sin(3x) \cos(x) \: \: \: \: \: \: \: \: \: \: \: \binom{ \sin(x) + \sin(y) = 2 \sin( \frac{x + y}{2} ) \cos( \frac{x - y}{2} ) }{} \\ = 2 \sin(3x) ( \cos(2x) + \cos(x) ) \\ = 2 \sin(3x) (2 \cos( \frac{3x}{2} ) \cos( \frac{x}{2} ) ) \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \binom{ \cos(x) + \cos(y) = 2 \cos( \frac{x + y}{2} ) \cos( \frac{x - y}{2} ) }{} \\ = 4 \cos( \frac{x}{2} ) \cos( \frac{3x}{2} ) \sin(3x)$

scroll left ☞