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# Find the value of $x$ if $sin \bigg ( sin^{-1} \frac{1}{5} + cos^{-1} \: x \bigg) = 1$.

Toolbox:
• $sin^{-1}x+cos^{-1}x=\large\frac{\pi}{2}$
• $sin\large\frac{\pi}{2}=1\: or \: sin^{-1}1=\large\frac{\pi}{2}$
Given $\; sin \bigg ( sin^{-1} \large\frac{1}{5} + cos^{-1} \: x \bigg) = 1$:

We know that $sin \large\frac{\pi}{2}=1 \Rightarrow$ the R.H.S of the equation $\:sin^{-1}\large\frac{1}{5}+cos^{-1}x$ should be =$\;\large\frac{\pi}{2}$

Given that $sin^{-1}x+cos^{-1}x=\large\frac{\pi}{2}$, if $sin^{-1}\large\frac{1}{5}+cos^{-1}x=\large\frac{\pi}{2} \Rightarrow x = \large\frac{\pi}{2}$

edited Mar 15, 2013