Prove that `( sin 5x - 2 sin 3x + sin x )/( cos 5x - cos x) = tan x`

Explanation

Here, we will use the following rules,

`sinA - sinB = 2cos((A+B)/2)sin((A-B)/2)`

`cosA-cosB = -2sin((A+B)/2)sin((A-B)/2)`

`L.H.S. = (sin5x - 2sin3x+sinx)/(cos5x-cosx)`

`= ((sin5x - sin3x)-(sin3x-sinx))/(cos5x-cosx)`

`=(2cos4xsinx - 2cos2xsinx)/(-2sin3xsin2x)`

`=(sinx(cos4x-cos2x))/(-sin3xsin2x)`

`=(sinx(-2sin3xsinx))/(-sin3xsin2x)`

...