Question : Which of the following is the value of $\sqrt{\frac{1-\sin 45^{\circ}}{1+\sin 45^{\circ}}}$?
Option 1: $\cos 45^{\circ} - \tan 45^{\circ}$
Option 2: $\tan 45^{\circ} - \sec 45^{\circ}$
Option 3: $\tan 45^{\circ}$
Option 4: $\sec 45^{\circ} - \tan 45^{\circ}$
Correct Answer: $\sec 45^{\circ} - \tan 45^{\circ}$
Solution :
$\sqrt{\frac{1-\sin 45^{\circ}}{1+\sin 45^{\circ}}}$
= $\sqrt{\frac{(1-\sin 45^{\circ})^2}{(1+\sin 45^{\circ})(1-\sin 45^{\circ})}}$
= $\sqrt{\frac{(1-\sin 45^{\circ})^2}{(1-\sin^2 45^{\circ})}}$
= $\sqrt{\frac{(1-\sin 45^{\circ})^2}{\cos^2 45^{\circ}}}$
= $\frac{1-\sin 45^{\circ}}{\cos 45^{\circ}}$
= $\sec 45^{\circ} - \tan 45^{\circ}$
Hence, the correct answer is $\sec 45^{\circ} - \tan 45^{\circ}$.
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