A) \[2\,\cos ec\,\theta \]
B) \[\frac{2\,\sin \theta }{\sqrt{\sec \theta }}\]
C) \[2\,\cos \theta \]
D) \[2\,sec\theta \]
Correct Answer: A
Solution :
\[\sqrt{\frac{\sec \theta -1}{\sec \theta +1}}+\sqrt{\frac{\sec \theta +1}{sec\theta -1}}\] \[=\sqrt{\frac{1-\cos \theta }{1+\cos \theta }}+\sqrt{\frac{1+\cos \theta }{1-\cos \theta }}\] \[=\sqrt{\frac{(1-\cos \theta )\,(1-\cos \theta )}{(1+\cos \theta )\,(1-\cos \theta )}}+\sqrt{\frac{(1+\cos \theta )+(1+\cos \theta )}{(1-\cos \theta \,(1+\cos \theta )}}\]\[=\frac{1-\cos \theta }{\sqrt{1-{{\cos }^{2}}\theta }}+\frac{1+\cos \theta }{\sqrt{1-{{\cos }^{2}}\theta }}\] \[=\frac{1-\cos \theta }{\sin \theta }+\frac{1+\cos \theta }{\sin \theta }=\frac{2}{\sin \theta }=2\cos ec\theta \]
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