trig identities

Question

tanx/(1+cosx)+sinx/(1-cosx)=cotx+secxcscx

Verify the identity.

Answer 1

Left hand side identity

= Tanx/(1+Cosx)+Sinx/(1-Cosx)

= [Tanx(1-Cosx)+Sinx(1+Cosx)]/[(1+Cosx)(1-Cosx)]

= [Tanx-TanxCosx+Siinx+SinxCosx]/(1-Cos^2x)

We know that Tanx = Sinx/Cosx.

= [Tanx-Sinx+Sinx+SinxCosx]/Sin^2x

= [Tanx+SinxCosx]/Sin^2x

= Tanx/Sin^2x + (SinxCosx)/Sin^2x

= 1/CosxSinx + Cosx/Sinx

= SecxCscx + Cotx

= Right hand side identity.