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# Find the value of $\cos[2\cos^{-1}x+\sin^{-1}x]$ at $x=\large\frac{1}{5}$ where $0\leq \cos^{-1} x\leq \pi$ and $\large\frac{-\pi}{2}$$\leq \sin^{-1}x\leq \large\frac{\pi}{2} (a)\;\large\frac{2\sqrt 6}{5}$$\qquad(b)\;\large\frac{2}{5}$$\qquad(c)\;-\large\frac{2\sqrt 6}{5}$$\qquad(d)\;0$

The above given interval indicate the principal value of $\cos^{-1}x$ and $\sin^{-1}x$ we have
$\cos[2\cos^{-1}x+\sin^{-1}x]$
We know that $cos^{-1}x+sin^{-1}x = \large\frac{\pi}{2}$
$\Rightarrow \cos[\cos^{-1}x+\large\frac{\pi}{2}]=-\sin(\cos^{-1}x)$
$\qquad\qquad\qquad\qquad\qquad=-\sin\sin^{-1}\sqrt{1-x^2}$
$\qquad\qquad\qquad\qquad\qquad=-\sqrt{1-x^2}$
$(at \:\:x=5),\qquad\qquad=-\sqrt{1-\large\frac{1}{25}}$
$\qquad\qquad\qquad\qquad\qquad=\large\frac{-2\sqrt 6}{5}$
Hence (c) is the correct answer.
edited Mar 26, 2014