Question

Prove that:

(cot θ − cosec θ)² = `(1 − cos θ)/(1 + cos θ)`

Answer 1

Taking LHS,

(cot θ − cosec θ)²

⇒ cot² θ + cosec² θ − 2 × cot θ × cosec θ

⇒ `(cos^2 θ)/(sin^2 θ) + 1/(sin^2 θ) - 2 xx (cos θ)/(sin θ) xx 1/sin θ`

⇒ `((1 + cos^2 θ))/(sin^2 θ) - (2cos θ)/sin^2 θ`

⇒ `((1 + cos^2 θ - 2 cos θ))/(sin^2 θ)`

⇒ `((1 - cos θ)^2)/sin^2 θ       ...["As" sin^2 θ = 1 - cos^ 2θ]`

⇒ `(1 - cos θ)^2/(1 - cos^2 θ)`

⇒ `(1 - cos θ)^2/[[(1 -cosθ)(1 + cos θ)]]`

⇒ `((1 - cos θ))/((1 + cos θ))`

LHS = RHS

Hence, proved.