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