Prove that!!!!!!!!!!!!!!!!!!!

Question

(1 - cosx / sinx) + (sinx / 1 - cosx) = 2 csc x?

Answer 1

The trigonometric equation is [(1 - cos x)/sin x] + [sin x/(1 - cos x)] = 2 csc x.

Left hand side identity : [(1 - cos x)/sin x] + [sin x/(1 - cos x)]

= [(1 - cos x)² + sin² x]/[(1 - cos x)sin x]

= [(1 - 2cos x + cos² x + sin² x]/[(1 - cos x)sin x]

Phythagoras identity : cos² x + sin² x = 1.

= [(1 - 2cos x + 1]/[(1 - cos x)sin x]

= [2 - 2cos x]/[(1 - cos x)sin x]

= 2[1 - cos x]/[(1 - cos x)sin x]

= 2/sin x

Using reciprocal identity : 1/sin x = csc x.

= 2csc x

= Right hand side identity.