The value of
Question:

The value of $\left(\cot \frac{x}{2}-\tan \frac{x}{2}\right)^{2}(1-2 \tan x \cot 2 x)$ is

(a) 1

(b) 2

(c) 3

(d) 4

Solution:

(d) 4

We have:

$\left(\cot \frac{x}{2}-\tan \frac{x}{2}\right)^{2}(1-2 \tan x \cot 2 x)$

$\left(\cot ^{2} \frac{x}{2}-2 \cot \frac{x}{2} \tan \frac{x}{2}+\tan ^{2} \frac{x}{2}\right)\left\{1-2 \tan x\left(\frac{\cot ^{2} x-1}{2 \cot x}\right)\right\}$

$\left(\cot ^{2} \frac{x}{2}-2+\tan ^{2} \frac{x}{2}\right)\left\{1-\tan x\left(\frac{\cot ^{2} x-1}{\cot x}\right)\right\}$

$\left(\cot ^{2} \frac{x}{2}+\tan ^{2} \frac{x}{2}-2\right)\left(1-\frac{\cot x-\tan x}{\cot x}\right)$

$\left(\cot ^{2} \frac{x}{2}+\tan ^{2} \frac{x}{2}-2\right)\left(\tan ^{2} x\right)$

$\left(\cot ^{2} \frac{x}{2}+\tan ^{2} \frac{x}{2}-2\right)\left(\frac{2 \tan \frac{x}{2}}{1-\tan ^{2} \frac{x}{2}}\right)^{2}$

$=\frac{1}{\left(1-\tan ^{2} \frac{x}{2}\right)^{2}}\left(4+4 \tan ^{4} \frac{x}{2}-8 \tan ^{2} \frac{x}{2}\right)$

$=\frac{1}{\left(1-\tan ^{2} \frac{x}{2}\right)^{2}}\left(4-8 \tan ^{2} \frac{x}{2}+4 \tan ^{4} \frac{x}{2}\right)$

$=\frac{4}{\left(1-\tan ^{2} \frac{x}{2}\right)^{2}}\left\{\left(\tan ^{2} \frac{x}{2}\right)^{2}-2\left(\tan ^{2} \frac{x}{2}\right)+1\right\}$

$=\frac{4\left(\tan ^{2} \frac{x}{2}-1\right)^{2}}{\left(1-\tan ^{2} \frac{x}{2}\right)^{2}}$

= 4

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