1 - \dfrac{9}{25}} = \dfrac{4}{5}\). Let \(\cos^{-1}\left(\dfrac{12}{13}\right) = \phi\). Then \(\cos\phi = \dfrac{12}{13}\) and \(0 < \phi < \dfrac{\pi}{2}\). \(\therefore \sin\phi = \sqrt{1 - \dfrac{144}{169}} = \dfrac{5}{13}\). Now \(\sin(\theta + \phi) = \sin\theta\cos\phi + \cos\theta\sin\phi = \left(\dfrac{3}{5}\right)\left(\dfrac{12}{13}\right) + \left(\dfrac{4}{5}\right)\left(\dfrac{5}{13}\right) = \dfrac{36}{65} + \dfrac{20}{65} = \dfrac{56}{65}\) ...(1) We must also confirm that \(\theta + \phi\) lies in the principal branch \(\left[-\dfrac{\pi}{2},\ \dfrac{\pi}{2}\right]\) of \(\sin^{-1}\). For that, compute \(\cos(\theta + \phi) = \cos\theta\cos\phi - \sin\theta\sin\phi = \left(\dfrac{4}{5}\right)\left(\dfrac{12}{13}\right) - \left(\dfrac{3}{5}\right)\left(\dfrac{5}{13}\right) = \dfrac{48}{65} - \dfrac{15}{65} = \dfrac{33}{65} > 0\). Since \(\theta + \phi\) is positive with positive cosine, \(0 < \theta + \phi < \dfrac{\pi}{2}\). \(\therefore\) from (1), \(\theta + \phi = \sin^{-1}\left(\dfrac{56}{65}\right)\). \(\therefore \sin^{-1}\left(\dfrac{3}{5}\right) + \cos^{-1}\left(\dfrac{12}{13}\right) = \sin^{-1}\left(\dfrac{56}{65}\right)\). Hence proved.26:T45f,Let \(\cos^{-1}\left(\dfrac{4}{5}\right) = \theta\). Then \(\cos\theta = \dfrac{4}{5}\) and, since \(\dfrac{4}{5} > 0\), \(0 < \theta < \dfrac{\pi}{2}\). \(\therefore \sin\theta = \sqrt{1 - \cos^{2}\theta} = \sqrt{1 - \dfrac{16}{25}} = \dfrac{3}{5}\) (positive, as \(\theta\) is in the first quadrant). Let \(\cos^{-1}\left(\dfrac{12}{13}\right) = \phi\). Then \(\cos\phi = \dfrac{12}{13}\) and \(0 < \phi < \dfrac{\pi}{2}\). \(\therefore \sin\phi = \sqrt{1 - \dfrac{144}{169}} = \dfrac{5}{13}\). Now \(\cos(\theta + \phi) = \cos\theta\cos\phi - \sin\theta\sin\phi = \left(\dfrac{4}{5}\right)\left(\dfrac{12}{13}\right) - \left(\dfrac{3}{5}\right)\left(\dfrac{5}{13}\right) = \dfrac{48}{65} - \dfrac{15}{65} = \dfrac{33}{65}\) ...(1) Also \(0 < \theta < \dfrac{\pi}{2}\) and \(0 < \phi < \dfrac{\pi}{2}\), so \(0 < \theta + \phi < \pi\), which

Trigonometric Functions

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