Question

please prove the following identity

Answer 1

Proof :

∴ L.H.S. = tan²θ / (secθ - 1)²

= (sin²θ / cos²θ) / (1/cosθ - 1)²

= (sin²θ / cos²θ) / {(1 - cosθ) / cosθ}²

= (sin²θ / cos²θ) / {(1 - cosθ)² / cos²θ}

= sin²θ / (1 - cosθ)²

= (1 - cos²θ) / (1 - cosθ)²

= {(1 + cosθ)(1 - cosθ)} / (1 - cosθ)²

= (1 + cosθ) / (1 - cosθ)

= {(1 + cosθ)(1 + cosθ)} / {(1 - cosθ)(1 + cosθ)}

= (1 + cosθ)² / (1 - cos²θ)

= {(1 + cosθ)/sinθ}²

= (1/sinθ + cosθ/sinθ)²

= (cosecθ + cotθ)² = R.H.S

Hence, proved.