# What does $cos^{-1} \frac {4}{5} +cos^{-1} \frac{12}{13}$ reduce to?

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