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# Find the simplified form of $cos^{-1}(\frac{3}{5}cos x+\frac{4}{5}sin x),$where $x\in (\frac{-3}{4},\frac{3}{4})$

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Toolbox:
• $cosA\: cosB+sinA\: sinB=cos(A-B)$
• $sinx=\sqrt{1-cos^2x}$
Let $\frac{3}{5}=cos\theta$
$\Rightarrow\:sin\theta=\frac{4}{5}$
$\Rightarrow\:\frac{3}{5}cosx-\frac{4}{5}sinx=cos\theta cosx+sin\theta sinx$
By taking A=$\theta$ and B=x in the above formula
$cos\theta cosx+sin\theta sinx=cos(\theta-x)$
The given expression becomes
$cos^{-1} [cos\theta cosx+sin\theta sinx]$
$= cos^{-1}\: cos(\theta-x)=\theta-x$
Substituting the value of $\theta$, we get
$=cos^{-1}\frac{3}{5}-x$
answered Mar 2, 2013
edited Mar 7, 2013