Question

If the $\theta$ lies in the second quadrant, then the value of $\sqrt{\left(\frac{1-\sin \theta }{1+\sin \theta }\right)}+\sqrt{\left(\frac{1+\sin \theta }{1-\sin \theta }\right)}$ is

• A. $2\sec \theta$
• B. $-2\sec \theta$
• C. $2\mathrm{cosec}\theta$
• D. none of these

Answer 1

Answer: (B)

$-2\sec \theta$

Consider the given expression.

$\frac{\sqrt{(1-\sin \theta )}}{\sqrt{(1+\sin \theta )}}+\frac{\sqrt{(1+\sin \theta )}}{\sqrt{(1-\sin \theta )}}$

$=\frac{(1-\sin \theta )+(1+\sin \theta )}{\sqrt{1^2-{\sin }^2\theta }}$

$=\frac{1-\sin \theta +1+\sin \theta }{\sqrt{1^2-{\sin }^2\theta }}$

$=\frac{2}{\sqrt{1-{\sin }^2\theta }}$

$=\frac{2}{\sqrt{{\cos }^2\theta }}$

$=\frac{2}{\cos \theta }$

$=2\times \frac{1}{\cos \theta }$ since, $\sec \theta =\frac{1}{\cos \theta }$

$=2\sec \theta$

If $\theta$ lies in second quadrant then the value

$=-2\sec \theta$

Hence , the value is $-2sec\theta$.