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

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

Explanation for the correct option:

Step 1. Find the value of $\cos \frac{(\theta -\varphi )}{2}$:

Given, $\cos \theta =\frac{3}{5}$ and $\cos \varphi =\frac{4}{5}$, where $\theta$ and $\varphi$ are positive acute angles

Thus, $\sin \theta =\frac{4}{5}$ and $\sin \varphi =\frac{3}{5}$

As we know,

$\mathit{c}\mathit{o}{\mathit{s}}^{\mathbf{2}}\frac{\mathbf{(}\mathbf{\theta }\mathbf{}\mathbf{–}\mathbf{}\mathit{\varphi }\mathbf{)}}{\mathbf{2}}\mathbf{}\mathbf{=}\mathbf{}\frac{\mathbf{[}\mathbf{1}\mathbf{}\mathbf{+}\mathbf{}\mathbf{c}\mathbf{o}\mathbf{s}\mathbf{(}\mathbf{\theta }\mathbf{}\mathbf{–}\mathbf{}\mathit{\varphi }\mathbf{\left)}\mathbf{\right]}}{\mathbf{2}}$ …(i)

$\mathit{c}\mathit{o}\mathit{s}\mathbf{(}\mathit{\theta }\mathbf{}\mathbf{–}\mathbf{}\mathit{\varphi }\mathbf{)}\mathbf{}\mathbf{=}\mathbf{}\mathit{c}\mathit{o}\mathit{s}\mathbf{}\mathit{\theta }\mathbf{}\mathit{c}\mathit{o}\mathit{s}\mathbf{}\mathit{\varphi }\mathbf{}\mathbf{+}\mathbf{}\mathit{s}\mathit{i}\mathit{n}\mathbf{}\mathit{\theta }\mathbf{}\mathit{s}\mathit{i}\mathit{n}\mathbf{}\mathit{\varphi }$ …(ii)

Step 2. Put the values of $\cos \theta ,\cos \varphi ,\sin \theta ,\sin \varphi$ in equation (ii), we get

$\begin{array}{rcl}\cos \left(\theta -\varphi \right)& =& \left(\frac{3}{5}\right)\left(\frac{4}{5}\right)+\left(\frac{4}{5}\right)\left(\frac{3}{5}\right)\\ & =& \left(\frac{12}{25}\right)+\left(\frac{12}{25}\right)\\ & =& \frac{24}{25}\end{array}$

Now, From (i), we get

$\begin{array}{rcl}{\cos }^2\frac{(\theta –\varphi )}{2}& =& \frac{\left[1+\left({\displaystyle \frac{24}{25}}\right)\right]}{2}\\ & =& \frac{49}{50}\end{array}$

$\begin{array}{rcl}\therefore \cos \frac{(\theta –\varphi )}{2}& =& \sqrt{\left(\frac{49}{50}\right)}\\ & =& \frac{\mathbf{7}}{\mathbf{5}\sqrt{\mathbf{2}}}\end{array}$

Hence, Option ‘B’ is Correct.