First slide

Multiple and sub- multiple Angles

Question

If $\tan x / 2 = cosec x - \sin x$ , then $\sec^{2} (x / 2) =$

Moderate

A

$\sqrt{5} + 1$
B

$\sqrt{5} - 1$
C

$\sqrt{5} - 2$
D

$\sqrt{5} + 2$

Solution

$\tan (x / 2) = \frac{1 + \tan^{2} (x / 2)}{2 \tan (x / 2)} - \frac{2 \tan (x / 2)}{1 + \tan^{2} (x / 2)}$
$\begin{array}{l} \Rightarrow 2 \tan^{2} (x / 2) (1 + \tan^{2} (x / 2) = {[1 + \tan^{2} (x / 2)]}^{2} - 4 \tan^{2} (x / 2) \\ \Rightarrow 2 \tan^{4} (x / 2) + 2 \tan^{2} (x / 2) = 1 + \tan^{4} (x / 2) - 2 \tan^{2} (x / 2) \\ \Rightarrow \tan^{4} (x / 2) + 4 \tan^{2} (x / 2) - 1 = 0 \\ \Rightarrow \tan^{2} (x / 2) = \sqrt{5} - 2 \Rightarrow \sec^{2} (x / 2) = \sqrt{5} - 1 \end{array}$

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