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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
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  • \( cos^{-1}x+cos^{-1}y=cos^{-1} (xy- \sqrt{1-x^2} \sqrt{1-y^2} )\)
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
answered Feb 13, 2013 by rvidyagovindarajan_1
edited Mar 19, 2013 by rvidyagovindarajan_1

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