Question

How do you verify `-cot(x/2) = (sin2x + sinx) / (cos2x - cosx)`?

Answer 1

Verify trig expression
`-cot (x/2) = (sin 2x + sin x)/(cos 2x - cos x)`

Explanation:

Use the trig identities to transform the right side:
sin 2x = sin x.cos x
`cos 2x = 2cos^2 x - 1`.
Transform the right side:
RS = `(sin 2x + sin x)/(cos 2x - cos x) = (sin x(2cos x + 1))/(2cos^2 x - cos x - 1` =

Factor the trinomial `(2cos^2 x - cos x - 1).`
Since (a + b + c = 0), use the Shortcut, the 2 factors are (cos x - 1) and (2cos x + 1). Finally,

`RS = (sin x(2cos x + 1))/((cos x - 1)(2cos x + 1)) = (sin x)/(cos x - 1)`
Since:
`sin x = 2sin (x/2).cos (x/2) and`
`(cos x - 1) = -2sin^2 (x/2)`, therefor
`RS = (2sin (x/2)(cos x/2))/(-2sin^2 (x/2)) =`
`= - cos (x/2)/(sin (x/2)) = -cot x/2`