Question : The value of $\frac{1+\sin A}{\cos A}+\frac{\cos A}{1+\sin A}$ is:
Option 1: $2 \sec A$
Option 2: $2 \operatorname{cosec} A$
Option 3: $ \sec A$
Option 4: $ \operatorname{cosec} A$
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Correct Answer: $2 \sec A$
Solution :
Consider $\frac{1+\sin A}{\cos A}+\frac{\cos A}{1+\sin A}$
= $\frac{(1+\sin A)(1+\sin A)+\cos^2A}{\cos A(1+\sin A)}$
= $\frac{1+\sin^2A+2\sin A + cos^2A}{\cos A(1+\sin A)}$
= $\frac{1+1+2\sin A}{\cos A(1+\sin A)}$ [Using $\sin^2A+\cos^2A=1$]
= $\frac{2+2\sin A}{\cos A(1+\sin A)}$
= $\frac{2(1+\sin A)}{\cos A(1+\sin A)}$
= $\frac{2}{\cos A}$
= $2\sec A$
Hence, the correct answer is $2\sec A$.
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