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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