Question

If 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. $noneofthese$

Answer 1

The correct option is A $2\sec \theta$

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

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

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

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

$=\frac{1-\sin \theta }{\cos \theta }+\frac{1+\sin \theta }{\cos \theta }$

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