Exhaustive value of x such that cos - 1 ( 8 x 1 + 16 x 2 ) = - 2 + 2 tan - 1 ( 4 x )

Question

Exhaustive value of $x$ such that $\cos^{- 1} (\frac{8 x}{1 + 16 x^{2}}) = - \frac{π}{2} + 2 \tan^{- 1} (4 x)$

Smart Achievers · Accepted Answer

(3)

$On solving \frac{π}{2} - \sin^{- 1} (\frac{2 \times 4 x}{1 + {(4 x)}^{2}}) = - \frac{π}{2} + 2 \tan^{- 1} (4 x)$

$\Rightarrow 2 \tan^{- 1} (4 x) = π - \sin^{- 1} (\frac{2 (4 x)}{1 + {(4 x)}^{2}})$

$4 x \geq 1; x \geq \frac{1}{4}$

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