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  4. If θ = cot-1 √( cos x) - tan-1 √( cos x) , then sin=θ

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Q. If $\theta$ = $\cot^{-1} \sqrt{\left(\cos x\right)}$ - $\tan^{-1} \sqrt{\left(\cos x\right)} ,$ then $\sin=\theta$

Inverse Trigonometric Functions

Solution:

$\theta = \tan^{-1} \left\{\frac{1}{\sqrt{\cos x}}\right\} - \tan^{-1} \sqrt{\cos x} $
$ = \tan^{-1} \left( \frac{\frac{1}{\sqrt{\cos x}} - \sqrt{\cos x}}{ 1+ \left\{\frac{1}{\sqrt{\cos x}} . \sqrt{\cos x}\right\}}\right) $
or $ \tan \theta = \frac{ 1 - \cos x}{2\sqrt{\cos x}}$
$ \therefore \tan\theta = \frac{ 1 -\cos x}{2 \sqrt{\cos x}} \therefore \sin^{2} \theta $
$ = \frac{1}{1 + \cot^{2} \theta} = \left(\frac{1-\cos x}{1+ \cos x}\right)^{2} = \tan^{4} \left(\frac{x}{2}\right) $
$ \Rightarrow \sin\theta = \tan^{2} \left(\frac{x}{2}\right) $