Question

If $A+B=\frac{\pi }{2}$, the maximum values of $\cos A\cos B$ is

• A. $\frac{1}{2}$
• B. $\frac{3}{4}$
• C. $1$
• D. $\frac{4}{3}$

Answer 1

The correct option is A

$\frac{1}{2}$

Explanation for the correct option:

Step 1. Find the maximum value of $\cos A\cos B$:

$\begin{array}{rcl}f\left(A\right)& =& \cos A\cos B\\ & =& \cos A\cos \left(\frac{\pi }{2}–A\right)\\ & =& \cos A\sin A\end{array}$

Differentiate it with respect to $A$:

$\begin{array}{rcl}f\text{'}\left(A\right)& =& {\cos }^{2}A–{\sin }^{2}A\\ & =& \cos 2A\end{array}$

Step 2. For maximum value, $f\text{'}\left(A\right)=0$

$\Rightarrow$$\cos 2A=0$

$\Rightarrow$ $2A=\frac{\pi }{2}$

$\Rightarrow$ $A=\frac{\pi }{4}$

Step 3. Differentiate $\begin{array}{rcl}& & f\text{'}\left(A\right)\end{array}$ again with respect to $A$:

$\begin{array}{rcl}f\text{'}\text{'}\left(A\right)& =& –2\sin 2A\\ & =& –2\sin \left(\frac{\pi }{2}\right)\\ & =& –2<0\end{array}$

$\therefore$$f\left(A\right)$ is maximum at $\frac{\pi }{4}$.

Thus, The maximum value of $\mathit{c}\mathit{o}\mathit{s}\left(\frac{\pi }{4}\right)\mathbf{}\mathit{s}\mathit{i}\mathit{n}\left(\frac{\pi }{4}\right)\mathbf{}\mathbf{=}\mathbf{}\frac{\mathbf{1}}{\mathbf{2}}$

Hence, Option ‘A’ is Correct.