Question

Prove the identity

$\sqrt{\frac{\cos ec\theta -1}{\cos ec\theta +1}}+\sqrt{\frac{\cos ec\theta +1}{\cos ec\theta -1}}=2sec\theta$

Answer 1

Prove the identity

Given data: $\sqrt{\frac{\cos ec\theta -1}{\cos ec\theta +1}}+\sqrt{\frac{\cos ec\theta +1}{\cos ec\theta -1}}=2sec\theta$

Proof:

L. H. S

$$\begin{gathered} =\sqrt{\frac{\cos ec\theta -1}{\cos ec\theta +1}}+\sqrt{\frac{\cos ec\theta +1}{\cos ec\theta -1}} \\ =\sqrt{\frac{\cos ec\theta -1}{\cos ec\theta +1}\times \frac{\cos ec\theta +1}{\cos ec\theta +1}}+\sqrt{\frac{\cos ec\theta +1}{\cos ec\theta -1}\times \frac{\cos ec\theta -1}{\cos ec\theta -1}} \\ =\sqrt{\frac{\cos e{c}^2\theta -1}{{(\cos ec\theta +1)}^2}}+\sqrt{\frac{\cos e{c}^2\theta -1}{{(\cos ec\theta -1)}^2}} \\ =\sqrt{\frac{\mathrm{co}{t}^2\theta }{{(\cos ec\theta +1)}^2}}+\sqrt{\frac{\mathrm{co}{t}^2\theta }{{(\cos ec\theta -1)}^2}} \\ =\frac{cot\theta }{\cos ec\theta +1}+\frac{cot\theta }{\cos ec\theta -1} \\ =cot\theta \left[\frac{1}{\cos ec\theta +1}+\frac{1}{\cos ec\theta -1}\right] \\ =cot\theta \left[\frac{\cos ec\theta -1+\cos ec\theta +1}{\cos e{c}^2\theta -1}\right] \\ =cot\theta \times \frac{2\cos ec\theta }{co{t}^2\theta } \\ =2\times \frac{1}{\sin \theta }\times \frac{1}{{\displaystyle \raisebox{1ex}{$\cos \theta $}\!\left/ \!\raisebox{-1ex}{$\sin \theta $}\right.}} \\ =2\times \frac{1}{\sin \theta }\times \frac{\sin \theta }{\cos \theta } \\ =\frac{2}{\cos \theta } \\ =2sec\theta \end{gathered}$$

= R. H. S

Therefore, L. H. S = R. H. S

Hence, $\sqrt{\frac{\cos ec\theta -1}{\cos ec\theta +1}}+\sqrt{\frac{\cos ec\theta +1}{\cos ec\theta -1}}=2sec\theta$ is prove.