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