Prove that `cos^(-1)(4/5)cos^(-1)((12)/(13))=cos^(-1)((33)/(65))`

Explanation

Let `\cos ^{-1} \frac{4}{5}=x`. Then, `\cos x=\frac{4}{5} \Rightarrow \sin x=\sqrt{1-(\frac{4}{5})^{2}}=\frac{3}{5}`


$$ \begin{aligned} &\therefore \tan x=\frac{3}{4} \Rightarrow x=\tan ^{-1} \frac{3}{4} \\ &\therefore \cos ^{-1} \frac{4}{5}-\tan ^{-1} \frac{3}{4} \end{aligned} $$
Now let `\cos ^{\wedge}(-1) \frac{12}{13}=y` Then `\cos y=\frac{12}{13} \Rightarrow \sin y=\frac{5}{13}`
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