If f ( x ) = cot x 1 + cot x and α + β = 5 π 4 then the value of f ( α ) f ( β ) =

Given, $f\left(x\right)=\frac{\cot x}{1+\cot x}$ and $\alpha +\beta =\frac{5\pi }{4}$

$\Rightarrow \cot \left(\mathrm{\alpha }+\mathrm{\beta }\right)=\cot \frac{5\mathrm{\pi }}{4}$

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

$\Rightarrow \mathrm{cot\alpha cot\beta }=1+\mathrm{cot\beta }+\mathrm{cot\alpha }$

Now, $f\left(\alpha \right)=\frac{\mathrm{cot\alpha }}{1+\cot \alpha }, f\left(\beta \right)=\frac{\mathrm{cot\beta }}{1+\mathrm{cot\beta }}$

Then, $f\left(\alpha \right)f\left(\beta \right)=\frac{\mathrm{cot\alpha }}{1+\cot \alpha }\times \frac{\mathrm{cot\beta }}{1+\mathrm{cot\beta }}$

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

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

$=\frac{1}{2}$