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Q.
Which of the following, satisfy the equation $2 \cos ^{-1} x=\cot ^{-1}\left(\frac{2 x^2-1}{\sqrt{4 x^2-4 x^4}}\right)$
Inverse Trigonometric Functions
Solution:
$2 \cdot \cos ^{-1} x=\cot ^{-1}\left(\frac{2 x^2-1}{2|x| \sqrt{1-x^2}}\right)$
Let $\cos ^{-1} x =\theta$
$2 \theta=\cot ^{-1}\left(\frac{\cos 2 \theta}{2|\cos \theta| \sin \theta}\right)$
Case I : If $\cos \theta>0, x >0 \Rightarrow 0< x <1$ then
$2 \theta=\cot ^{-1} \cot 2 \theta=2 \theta$ (identity)
Case II : $\cos \theta<0$, which not satisfy the equation