prove: 1-cosx/1+cosx = (cscx + cotx)^2

Question

Prove the identity...verify the identity.

Answer 1

Considering the trigonometry identity is (1 - Cos x)/(1 + Cos x) = (Csc x + Cot x)^(- 2).

Left hand side identity : (1 - Cos x)/(1 + Cos x).

Multiply the numerator and the denominator by (1 + cos x).

= [(1 - Cos x) * (1 + cos x)] / (1 + Cos x)^2

= (1 - cos^2 x) / (1 + Cos x)^2

pythagorean identity : sin^2 x + cos^2 x = 1.

= (sin^2 x) / (1 + Cos x)^2

= [(sin x) / (1 + Cos x)]^2

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

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

Reciprocal identity : csc x = 1/sin x.

= [(csc x) + (Cos x/sin x)]^(- 2)

Trigonometric identity : cot x = cos x / sin x.

= [csc x + Cot x]^(- 2)

= Right hand side identity.

Hence proved.