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If $\cos \theta = \frac{3}{5}$ and $\cos \phi = \frac{4}{5},$ where $\theta $ and $\phi $ are positive acute angles, then $\cos \frac{{\theta - \phi }}{2} = $
$\frac{7}{{\sqrt 2 }}$
$\frac{7}{{5\sqrt 2 }}$
$\frac{7}{{\sqrt 5 }}$
$\frac{7}{{2\sqrt 5 }}$
Solution
(b) We have $\cos \theta = \frac{3}{5}$ and $\cos \phi = \frac{4}{5}$.
Therefore $\cos (\theta – \phi ) = \cos \theta \cos \phi + \sin \theta \sin \phi $
$ = \frac{3}{5}.\frac{4}{5} + \frac{4}{5}.\frac{3}{5} = \frac{{24}}{{25}}$
But $2{\cos ^2}\left( {\frac{{\theta – \phi }}{2}} \right) = 1 + \cos (\theta – \phi ) = 1 + \frac{{24}}{{25}}= \frac{{49}}{{50}}$
$\therefore$ ${\cos ^2}\left( {\frac{{\theta – \phi }}{2}} \right) = \frac{{49}}{{50}}$.
Hence, $\cos \left( {\frac{{\theta – \varphi }}{2}} \right) = \frac{7}{{5\sqrt 2 }}$.
