Question

Prove :

(1 - cos θ)(1 + cos θ)(1 + cot2θ)  =  1

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Answer 1

Solution :

Let A = (1 - cos θ)(1 + cos θ)(1 + cot2θ) = 1 and B = 1.

A = (1 - cos θ)(1 + cos θ)(1 + cot2θ)

A = (1 - cos2θ)(1 + cot2θ)

Because sin2θ + cos2θ = 1, we have

sin2θ = 1 - cos2θ

Then,

A = sin2θ ⋅ (1 + cot2θ)

A = sin2θ + sin2θ ⋅ cot2θ

A = sin2θ + sin2θ ⋅ (cos2θ/sin2θ)

A = sin2θ + cos2θ

A = 1

A = B (Proved)