question_answer

If \[2\sec 2\alpha =\tan \beta +\cot \beta ,\]then one of the values of \[\alpha +\beta \]is [Karnataka CET 2000]

A) \[\frac{\pi }{4}\]

B) \[\frac{\pi }{2}\]

C) \[\pi \]

D) \[2\pi \]

Correct Answer: A

Solution :

The given equation may be written as \[\frac{2}{\cos 2\alpha }=\frac{\sin \beta }{\cos \beta }+\frac{\cos \beta }{\sin \beta }=\frac{{{\sin }^{2}}\beta +{{\cos }^{2}}\beta }{\cos \beta \sin \beta }\]\[=\frac{1}{\cos \beta .\sin \beta }\] Þ \[\cos 2\alpha =\sin 2\beta \] Þ \[\cos 2\alpha \]= \[\cos \,\left( \frac{\pi }{2}-2\beta  \right)\]Þ \[2\alpha =\frac{\pi }{2}-2\beta \]  Þ  Þ .