Question

Let, $f\left(\theta \right)=\frac{\cot \theta }{1+\cot \theta }$ and $\alpha +\beta =\frac{5\pi }{4}$, then the value of $f\left(\alpha \right).f\left(\beta \right)$ is

• A. $\frac{1}{2}$
• B. $-\frac{1}{2}$
• C. $2$
• D. None of these

Answer 1

The correct option is A $\frac{1}{2}$

We have, $f\left(\theta \right)=\frac{\cot \theta }{1+\cot \theta }$

$\alpha +\beta =\frac{5\pi }{4}$

$\cot (\alpha +\beta )=\frac{5\pi }{4}$

$\frac{\cot \alpha \cot \beta -1}{\cot \beta +\cot \alpha }=1$

$\Rightarrow \cot \alpha \cot \beta =\cot \beta +\cot \alpha +1$

Now,$f\left(\alpha \right)f\left(\beta \right)$

$\Rightarrow \frac{\cot \alpha }{1+\cot \alpha }\times \frac{\cot \beta }{1+\cot \beta }$

$\Rightarrow \frac{\cot \alpha \cot \beta }{1+\cot \alpha \cot \beta +\cot \alpha +\cot \beta }$

$=\frac{\cot \alpha \cot \beta }{2\cot \alpha \cot \beta }$

$=\frac{1}{2}$