(1+cos⁡x)/sin⁡x + sin⁡x/(1+cos⁡x ) = 2/sin⁡x

Question

I just don't understand how to do this...

Answer 1

Left hand side identity =

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

= [(1 + Cosx )^2 + Sin^2x ] / Sinx (1 + Cosx )

= [1 + Cos^2x + 2Cosx + Sin^2x ] / Sinx (1 + Cosx )

= [1 + 2Cosx + Cos^2x + Sin^2x ] / Sinx (1 + Cosx )

We know that Sin^2x + Cos^2x = 1

= [1 + 2Cosx +1] / Sinx (1 + Cosx )

= [2 + 2Cosx ] / Sinx (1 + Cosx )

= 2(1 + Cosx ) / Sinx (1 + Cosx )

= 2/Sinx

= Right hand side identity