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

Answer 1

The correct option is B

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

We have cos $\theta$ = $\frac{3}{5}$ and cos $\varphi$ = $\frac{4}{5}$.

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

= $\frac{3}{5}$ . $\frac{4}{5}$ + $\frac{4}{5}$ . $\frac{3}{5}$ = $\frac{24}{25}$

But $2co{s}^2\left(\frac{\theta -\varphi }{2}\right)=1+cos(\theta -\varphi )$ = 1 + $\frac{24}{25}$ = $\frac{49}{50}$

∴ $co{s}^2\left(\frac{\theta -\varphi }{2}\right)$ = $\frac{49}{50}$. Hence, $cos\left(\frac{\theta -\varphi }{2}\right)$ = $\frac{7}{5\sqrt{2}}$.