Question

Prove the following trigonometric identities.

(i) $\sqrt{\frac{\sec \theta -1}{\sec \theta +1}}+\sqrt{\frac{\sec \theta +1}{\sec \theta -1}}=2\mathrm{cosec}\theta$
(ii) $\sqrt{\frac{1+\sin \theta }{1-\sin \theta }}+\sqrt{\frac{1-\sin \theta }{1+\sin \theta }}=2\sec \theta$
(iii) $\sqrt{\frac{1+\cos \theta }{1-\cos \theta }}+\sqrt{\frac{1-\cos \theta }{1+\cos \theta }}=2\mathrm{cosec}\theta$
(iv) $\frac{\sec \theta -1}{\sec \theta +1}={\left(\frac{\sin \theta }{1+\cos \theta }\right)}^2$
(v) $\frac{\sin \theta +1-\cos \theta }{\cos \theta -1+\sin \theta }=\frac{1+\sin \theta }{\cos \theta }$

Answer 1

(i) We have,

(ii) We have,

(iii) We have,

(iv) We have,

Multiplying both the numerator and the denominator by, we have

(v) We have,

Multiplying both the numerator and the denominator by , we have