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(A) Let $x = \cos^{-1} \frac{1}{3} + \cos^{-1} \frac{1}{5}$ and $y = \sin^{-1} \frac{1}{3} + \sin^{-1} \frac{1}{5}$.We know that $\cos^{-1} z = \frac{

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(A) Let $x = \cos^{-1} \frac{1}{3} + \cos^{-1} \frac{1}{5}$ and $y = \sin^{-1} \frac{1}{3} + \sin^{-1} \frac{1}{5}$.We know that $\cos^{-1} z = \frac{\pi}{2} - \sin^{-1} z$.Thus,$x = (\frac{\pi}{2} - \sin^{-1} \frac{1}{3}) + (\frac{\pi}{2} - \sin^{-1} \frac{1}{5}) = \pi - (\sin^{-1} \frac{1}{3} + \sin^{-1} \frac{1}{5}) = \pi - y$.Therefore,$\cos(x) = \cos(\pi - y) = -\cos(y)$.This implies $\cos(x) + \cos(y) = -\cos(y) + \cos(y) = 0$.

$\cos \left(\cos ^{-1} \frac{1}{3}+\cos ^{-1} \frac{1}{5}\right)+\cos \left(\sin ^{-1} \frac{1}{3}+\sin ^{-1} \frac{1}{5}\right) =$ . . . . . . .

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