Prove that: `(sin5x-2sin3x+sin x)/(cos5x-cos x)=tanx`

Explanation

`sin C-sin D=2 cos (frac{C+D}{2}) sin (frac{C-D}{2})`


`cos C-cos D=-2 sin (frac{C+D}{2}) sin (frac{C-D}{2})`


`{LHS}=frac{sin 5 x-2 sin 3 x+sin x}{cos 5 x-cos x}`


`=frac{(sin 5 x-sin 3 x)-(sin 3 x-sin x)}{cos 5 x-cos x}`


`=frac{2 cos 4 x sin x-2 cos 2 x sin x}{-2 sin 3 x sin 2 x}`


`=frac{2 sin x(cos 4 x-cos 2 x)}{-2 sin 3 x sin 2 x}`


`=frac{2 sin x(-2 sin 3 x sin x)}{-2 sin 3 x(2 sin x cos x)}`


`=frac{sin x}{cos x}`


...