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Question: If tan(cos<sup>–1</sup> x) = sin\(\left( \cot^{- 1}\frac{1}{2} \right)\), then x =...

If tan(cos–1 x) = sin(cot⁡−112)\left( \cot^{- 1}\frac{1}{2} \right), then x =

A

±53\pm \frac{5}{3}

B

±53\pm \frac{\sqrt{5}}{3}

C

±53\pm \frac{5}{\sqrt{3}}

D

None of these

Answer

±53\pm \frac{\sqrt{5}}{3}

Explanation

Solution

y' = 4(sin⁡–1x)31–x2\frac{4(\sin^{–1}x)^{3}}{\sqrt{1–x^{2}}} – 4(cos⁡–1x)31–x2\frac{4(\cos^{–1}x)^{3}}{\sqrt{1–x^{2}}} = 0

Ž sin–1 x = cos–1 x Ž x=12\begin{matrix} x = \frac{1}{\sqrt{2}} \end{matrix}

f(1) = π416\frac{\pi^{4}}{16}

f(–1) = π416\frac{\pi^{4}}{16} + p4 = 17π416\frac{17\pi^{4}}{16} ® maximum

f(1/2\sqrt{2}) = π4128\frac{\pi^{4}}{128} ® minimum

maximum + minimum = 17π416\frac{17\pi^{4}}{16} + π4128\frac{\pi^{4}}{128}= 137π4128\frac{137\pi^{4}}{128}