Question : What is the value of the expression: $\sin A(1+\frac{\sin A}{\cos A})+\cos A(1+\frac{\cos A}{\sin A})$?
Option 1: $\sec A+\operatorname{cosec}A$
Option 2: $\sin \mathrm{A}+\cos \mathrm{A}$
Option 3: $\sin \mathrm{A}-\cos \mathrm{A}$
Option 4: $\sec \mathrm{A}-\operatorname{cosec} \mathrm{A}$
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Correct Answer: $\sec A+\operatorname{cosec}A$
Solution :
Given:
$\sin A(1+\frac{\sin A}{\cos A})+\cos A(1+\frac{\cos A}{\sin A})$
= $\sin A(1+\tan A)+\cos A(1+\cot A)$
= $\sin A(1+\tan A)+\frac{\cos A}{\tan A}(1+\tan A)$
= $(1+\tan A)(\sin A +\frac{\cos^2 A}{\sin A})$
= $(1+\tan A)(\frac{\sin^2 A +cos^2 A}{\sin A})$
= $\frac{(1+\tan A)}{\sin A}$
= $\frac{(1+\frac{\sin A}{\cos A})}{\sin A}$
= $\frac{1}{\sin A} + \frac{1}{cos A}$
= $\operatorname{cosec} A+\sec A$ = $\sec A+\operatorname{cosec}A$
Hence, the correct answer is $\sec A+\operatorname{cosec}A$.
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