Question : Find the value of $\frac{\cos A}{1+\sin A}+\frac{1+\sin A}{\cos A}+2$.
Option 1: $1+2 \sec A$
Option 2: $2-\sec A$
Option 3: $2+\sec A$
Option 4: $2(1+\sec A)$
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Correct Answer: $2(1+\sec A)$
Solution :
Given:
$\frac{\cos A}{1+\sin A}+\frac{1+\sin A}{\cos A}+2$
= $\frac{\cos^2A+1+\sin A+\sin A+\sin^2A}{(1+\sin A)(\cos A)}+2$
= $\frac{2+2\sin A}{(1+\sin A)(\cos A)}+2$
= $\frac{2(1+\sin A)}{(1+\sin A)(\cos A)}+2$
= $\frac{2}{\cos A}+2$
= $2\sec A+2$
= $2(1+\sec A)$
Hence, the correct answer is $2(1+\sec A)$.
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