Question

prove that sin5(x) - 2sin3(x) + sin(x)/ cos5(x) - cos(x) = tanx

Answer 1

prove that $\sin 5(x) - 2\sin 3(x) + \sin(x)/\cos 5(x) - \cos(x) = \tan x$

$$\frac {\sin (5 x) - 2 \sin (3 x) + \sin (x)}{\cos (5 x) - \cos (x)} = \tan (x)$$ $$\begin{array}{l} \frac {\sin (5 x) - 2 \sin (3 x) + \sin (x)}{\cos (5 x) - \cos (x)} = \frac {2 \sin (3 x) \cos (2 x) - 2 \sin (3 x)}{- 2 \sin (3 x) \sin (2 x)} = \frac {2 \sin (3 x) (\cos (2 x) - 1)}{- 2 \sin (3 x) \sin (2 x)} = \\ = \frac {1 - \cos (2 x)}{\sin (2 x)} = \frac {\cos^ {2} x + \sin^ {2} x - \cos^ {2} x + \sin^ {2} x}{2 \sin x \cos x} = \frac {2 \sin^ {2} x}{2 \sin x \cos x} = \frac {\sin x}{\cos x} = \tan x \\ \end{array}$$

$\tan x = \tan x$

Proved!