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# If $x,y,z \in [-1,1]\;$ such that $\;\sin^{-1}x+\sin^{-1}y+\sin^{-1}z=\frac{3\pi}{2},$ find the value of $x^{2006}+y^{2007}+z^{2008}-\Large \frac{9}{x^{2006}+y^{2007}+z^{2008}}$

$\begin{array}{1 1} 0 \\ 3 \\ 6 \\ 9 \end{array}$

Toolbox:
• Principal interval of $sin^{-1}x \:is\:[-\large\frac{\pi}{2},\frac{\pi}{2}]$
• $sin\large\frac{\pi}{2}=1$
Ans = 0

Max value of $sin^{-1}x \: is \:\large \frac{\pi}{2}$ and min value is $\large\frac{-\pi}{2}$
If $sin^{-1}x+sin^{-1}y+sin^{-1}z=\large\frac{3\pi}{2}$ then $sin^{-1}x=\large\frac{\pi}{2}, \: sin^{-1}y=\large\frac{\pi}{2},\: sin^{-1}z=\large\frac{\pi}{2}$

therefore $x=y=z=sin\large\frac{\pi}{2}=1$
Substituting the values of x,y,z in the given expression we get

Ans $1+1+1-\frac{9}{3}=0$

edited Mar 19, 2013