Prove the following trigonometric identities.(i) sec θ-1sec θ+1+sec θ+1sec θ-1=2 cosec θ(ii) 1+sin θ1-sin θ+1-sin θ1+sin θ=2 sec θ(iii) 1+cos θ1-cos θ+1-cos θ1+cos θ=2 cosec θ(iv) sec θ-1sec θ+1=sin θ1+cos θ2(v) sinθ+1-cosθcosθ-1+sinθ=1+sinθcosθ

Prove the following trigonometric identities.(i) sec θ-1sec θ+1+sec

1.	Prove the following trigonometric identities.(i) sec θ-1sec θ+1+sec θ+1sec θ-1=2 cosec θ(ii) 1+sin θ1-sin θ+1-sin θ1+sin θ=2 sec θ(iii) 1+cos θ1-cos θ+1-cos θ1+cos θ=2 cosec θ(iv) sec θ-1sec θ+1=sin θ1+cos θ2(v) sinθ+1-cosθcosθ-1+sinθ=1+sinθcosθ
Answer» Prove the following trigonometric identities. (i) $\sqrt{\frac{\sec θ - 1}{\sec θ + 1}} + \sqrt{\frac{\sec θ + 1}{\sec θ - 1}} = 2 cosec θ$ (ii) $\sqrt{\frac{1 + \sin θ}{1 - \sin θ}} + \sqrt{\frac{1 - \sin θ}{1 + \sin θ}} = 2 \sec θ$ (iii) $\sqrt{\frac{1 + \cos θ}{1 - \cos θ}} + \sqrt{\frac{1 - \cos θ}{1 + \cos θ}} = 2 cosec θ$ (iv) $\frac{\sec θ - 1}{\sec θ + 1} = {(\frac{\sin θ}{1 + \cos θ})}^{2}$ (v) $\frac{\sin θ + 1 - \cos θ}{\cos θ - 1 + \sin θ} = \frac{1 + \sin θ}{\cos θ}$