1 / tanx (1 + cos2x) = csc2x

Question

I am trying to prove this and I am really not sure how to start. I know I am supposed to use half-angle identities, but I'm not sure which one.

Answer 1

Recall the formula tanA = sinA/cosA

sin2A = 2sinAcosA

cos2A = cos^2A-sin^2A

sinA = 1/cscA

Left hand side =1/tanx(1+cos2x)

= 1/(sinx/cosx)(1+cos^2x-sin^2x)

= 1/(sinx/cosx(cos^2x+cos^2x))

= 1/(sinx/cosx(2cos^2x)

= 1/2sinxcosx

= 1/sin2x

= csc2x

= Right hand side