Question

PLEASE PROVE THIS
STEP BY STEP

Answer 1

To prove :-

$\frac{ \sin(5x) - 2 \sin(3x) + \sin(x) }{ \cos(5x) - \cos(x) } = \tan(x)$

Formula used

$\sin( \alpha ) + \sin( \beta ) = 2 \sin( \frac{ \alpha + \beta }{2} ) \cos(\frac{ \alpha - \beta }{2}) \\ \\ \sin( \alpha ) - \sin( \beta ) = 2 \sin( \frac{ \alpha - \beta }{2} ) \cos(\frac{ \alpha + \beta }{2}) \\ \\ \cos( \alpha ) + \cos( \beta ) = 2 \cos( \: \frac{ \alpha + \beta }{2}) \cos(\frac{ \alpha + \beta }{2}) \\ \\ \cos( \beta ) - \cos( \alpha ) = 2 \sin(\frac{ \alpha + \beta }{2}) \sin(\frac{ \alpha - \beta }{2})$

Solutions

$\frac{ \sin(5x) - 2 \sin(3x) + \sin(x) }{ \cos(5x) - \cos(x) } \\ \\ = \frac{ \: \sin(5x) + \sin(x) - 2 \sin(3x) }{ \cos(5x) - \cos(x)} \\ \\ = \frac{2sin3x \cos2x - 2 \sin3x }{ - 2 \sin3x \: \sin2x } \\ \\ = \frac{2 \sin3x \: ( \cos2x - 1) }{ \: \:- 2 \sin3x \: \sin2x} \\ \\ = \frac{ - 2 { \sin }^{2}x }{ - 2 \sin(x) \cos(x) }$

= tanx