Question

The maximum value of $\cos {\alpha }_1\cdot \cos {\alpha }_2\cdot \cos {\alpha }_3...\cos {\alpha }_n$ under the restriction $0\le {\alpha }_1,{\alpha }_2,...,{\alpha }_n\le \pi /2$ and $\cot {\alpha }_1=\cot {\alpha }_2=....\cot {\alpha }_n=1$ is :

• A. $1/2^{n/2}$
• B. $1/2^n$
• C. $1/2^{n-1}$
• D. $1$

Answer 1

The correct option is A $1/2^{n/2}$

$\cot {\alpha }_1=\cot {\alpha }_2....=\cot {\alpha }_n=1$
where $0\le {\alpha }_i\le \pi /2i=1,2=-n$
$\Rightarrow {\alpha }_1={\alpha }_2...={\alpha }_n=\pi /4$
$\therefore$ max. value of $\cos {\alpha }_1\cdot \cos {\alpha }_2....\cos {\alpha }_n$
$={\left(\frac{1}{\sqrt{2}}\right)}^n=\frac{1}{(2{)}^{n/2}}$