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Question
The value of \[\left( \cot \frac{x}{2} - \tan \frac{x}{2} \right)^2 \left( 1 - 2 \tan x \cot 2 x \right)\] is
Options
1
2
3
4
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Solution
4
\[\text{ We have } , \]
\[ \left( \cot\frac{x}{2} - \tan\frac{x}{2} \right)^2 \left( 1 - 2\text{ tan } x \cot2x \right)\]
\[\left( \cot^2 \frac{x}{2} - 2\cot\frac{x}{2}\tan\frac{x}{2} + \tan^2 \frac{x}{2} \right) \left\{ 1 - 2\text{ tan } x \left( \frac{\cot^2 x - 1}{2\text{ cot } x} \right) \right\}\]
\[\left( \cot^2 \frac{x}{2} - 2 + \tan^2 \frac{x}{2} \right)\left\{ 1 - \text{ tan } x \left( \frac{\cot^2 x - 1}{\text{ cot } x} \right) \right\}\]
\[\left( \cot^2 \frac{x}{2} + \tan^2 \frac{x}{2} - 2 \right)\left( 1 - \frac{\text{ cot } x - \text{ tan } x}{\text{ cot } x} \right)\]
\[\left( \cot^2 \frac{x}{2} + \tan^2 \frac{x}{2} - 2 \right)\left( \tan^2 x \right)\]
\[\left( \cot^2 \frac{x}{2} + \tan^2 \frac{x}{2} - 2 \right) \left( \frac{2\tan\frac{x}{2}}{1 - \tan^2 \frac{x}{2}} \right)^2\]
\[= \frac{1}{\left( 1 - \tan^2 \frac{x}{2} \right)^2}\left( 4 + 4 \tan^4 \frac{x}{2} - 8 \tan^2 \frac{x}{2} \right)\]
\[ = \frac{1}{\left( 1 - \tan^2 \frac{x}{2} \right)^2}\left( 4 - 8 \tan^2 \frac{x}{2} + 4 \tan^4 \frac{x}{2} \right)\]
\[ = \frac{4}{\left( 1 - \tan^2 \frac{x}{2} \right)^2} \left\{ \left( \tan^2 \frac{x}{2} \right)^2 - 2\left( \tan^2 \frac{x}{2} \right) + 1 \right\}\]
\[ = \frac{4 \left( \tan^2 \frac{x}{2} - 1 \right)^2}{\left( 1 - \tan^2 \frac{x}{2} \right)^2}\]
\[ = 4\]
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