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# Prove the following : $cos^{-1} \bigg( \frac{4}{5} \bigg)+ cos^{-1} \bigg( \frac{12}{13} \bigg)= cos^{-1} \bigg( \frac{33}{65} \bigg)$

This question is Q.No.5 of misc. chapter 2

Toolbox:
• $cos^{-1}x+cos^{-1}y=cos^{-1} (xy- \sqrt{1-x^2} \sqrt{1-y^2} )$
Given $cos^{-1} \large\frac {4}{5} +cos^{-1} \large\frac{12}{13}$
We know that $cos^{-1}x+cos^{-1}y=cos^{-1} (xy- \sqrt{1-x^2} \sqrt{1-y^2} )$
By taking $x=\large\frac{4}{5}\:and\:y=\large\frac{12}{13}$in the above formula, we get
$cos^{-1}\large\frac{4}{5}+cos^{-1}\large\frac{12}{13}=$ $cos^{-1} \bigg[ \large\frac{4}{5}.\large\frac{12}{13}-\sqrt{1-\large\frac{16}{25}} \sqrt{1-\large\frac{144}{169}} \bigg]$
$= cos^{-1} \bigg[ \large\frac{48}{65}-\large\frac{3}{5}.\large\frac{5}{13} \bigg]=\:cos^{-1}\big(\large\frac{48}{65}-\large\frac{15}{65}\big)$
$=cos^{-1} \large\frac{33}{65}$ = R.H.S
edited Mar 19, 2013