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# Prove that $cos^{-1} \frac {4}{5} +cos^{-1} \frac{12}{13}=cos^{-1} \frac{33}{65}$

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 15, 2013