Question

Verify the following with the help of identities.

(i) ((1 + cotθ + tanθ)(sinθ - cosθ))/(sec³θ - cosec³θ) = sin² θ cos²θ

(ii) (sinθ + cosθ)/(sinθ - cosθ) + (sinθ - cosθ)/(sinθ + cosθ) = 2/(1 - 2cos²θ) = 2/(2cos²θ - 1)

Answer 1

(i) LHS = ((1 + cotθ + tanθ)(sinθ - cosθ))/(sec³θ - cosec³θ)

= sin² θ cos²θ

= R.H.S.

(ii) LHS = (sinθ + cosθ)/(sinθ - cosθ) + (sinθ - cosθ)/(sinθ + cosθ)

= 2/(1 - 2cos²θ)

= R.H.S.

Again by equation (i)

= 2/(2cos²θ - 1)

= R.H.S.