Question

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

Answer 1

$$\begin{gathered} \mathrm{L}.\mathrm{H}.\mathrm{S}={\cos }^{-1}\frac{4}{5}+{\cos }^{-1}\frac{12}{13} \\ ={\cos }^{-1}\left[\frac{4}{5}\times \frac{12}{13}-\sqrt{1-{\left(\frac{4}{5}\right)}^2}\sqrt{1-{\left(\frac{12}{13}\right)}^2}\right]\left[\because {\cos }^{-1}x+{\cos }^{-1}y={\cos }^{-1}\left(xy-\sqrt{1-x^2}\sqrt{1-y^2}\right)\right] \\ ={\cos }^{-1}\left[\frac{48}{65}-\frac{3}{5}\times \frac{5}{13}\right] \\ ={\cos }^{-1}\left(\frac{48-15}{65}\right) \\ ={\cos }^{-1}\frac{33}{65}=\mathrm{R}.\mathrm{H}.\mathrm{S} \end{gathered}$$