Prove that

Question:

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

Solution:

L. H. S $=\cos ^{-1} \frac{4}{5}+\cos ^{-1} \frac{12}{13}$

$=\cos ^{-1}\left[\frac{4}{5} \times \frac{12}{13}-\sqrt{1-\left(\frac{4}{5}\right)^{2}} \sqrt{1-\left(\frac{12}{13}\right)^{2}}\right]$                   $\left[\because \cos ^{-1} x+\cos ^{-1} y=\cos ^{-1}\left(x y-\sqrt{1-x^{2}} \sqrt{1-y^{2}}\right)\right]$

$=\cos ^{-1}\left[\frac{48}{65}-\frac{3}{5} \times \frac{5}{13}\right]$

$=\cos ^{-1}\left(\frac{48-15}{65}\right)$

$=\cos ^{-1} \frac{33}{65}=$ R. H.S

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