Question

If $\cos \theta =\frac{3}{5}$ and $\cos \varphi =\frac{4}{5}$, where $\theta$ and $\varphi$ are positive acute angles,
then $\cos \frac{\theta -\varphi }{2}$ =

• A. $\frac{7}{2\sqrt{5}}$
• B. $\frac{7}{\sqrt{2}}$
• C. $\frac{7}{5\sqrt{2}}$
• D. $\frac{7}{\sqrt{5}}$

Answer 1

The correct option is C

$\frac{7}{5\sqrt{2}}$

We have
$\cos \theta =\frac{3}{5}$ and $\cos \varphi =\frac{4}{5}$
and $\theta$ and $\varphi$ are positive acute angles

$\Rightarrow \cos (\theta -\varphi )=\cos \theta \cos \varphi +\sin \theta sin\varphi$

$\Rightarrow \cos (\theta -\varphi )=\frac{3}{5}.\frac{4}{5}+\frac{4}{5}.\frac{3}{5}\Rightarrow \cos (\theta -\varphi )=\frac{24}{25}$

But
$2{\cos }^2\left(\frac{\theta -\varphi }{2}\right)=1+\cos (\theta -\varphi )$
$\Rightarrow 2{\cos }^2\left(\frac{\theta -\varphi }{2}\right)=1+\frac{24}{25}\Rightarrow 2{\cos }^2\left(\frac{\theta -\varphi }{2}\right)=\frac{49}{25}$

$\therefore {\cos }^2\left(\frac{\theta -\varphi }{2}\right)=\frac{49}{50}$.
Hence, $\cos \left(\frac{\theta -\varphi }{2}\right)=\frac{7}{5\sqrt{2}}$.
Hence the correct answer is Option B