Which expression is equal to $\sqrt{\dfrac{1+\sin A}{1-\sin A}}$ for an acute angle $A$?
A$\sec A-\tan A$
B$\cosec A+\cot A$
C$\sec A+\tan A$
D$\sin A+\cos A$
Answer & Solution
Correct answer: C. $\sec A+\tan A$
Underlying idea: rationalise the expression inside the square root using $1-\sin^2 A=\cos^2 A$.
$$\sqrt{\frac{1+\sin A}{1-\sin A}}=\sqrt{\frac{(1+\sin A)^2}{1-\sin^2 A}}=\sqrt{\frac{(1+\sin A)^2}{\cos^2 A}}.$$
Since $A$ is acute, $\cos A>0$, so this becomes
$$\frac{1+\sin A}{\cos A}=\frac{1}{\cos A}+\frac{\sin A}{\cos A}=\sec A+\tan A.$$
Option A is tempting because $(\sec A-\tan A)(\sec A+\tan A)=1$
so it is actually the reciprocal, not the given expression. Option B uses cosec and cot, which arise from dividing by $\sin A$, not $\cos A$.
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