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