Question

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

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

Answer 1

Answer: (A)

$\frac{1}{2}$

$f\left(\theta \right)=\frac{\cot \theta }{1+\cot \theta }=\frac{\cos \theta }{\cos \theta +\sin \theta }$
$\Rightarrow f\left(\theta \right)=\frac{\cos \theta }{\sqrt{2}(\cos \frac{\pi }{4}\cos \theta +\sin \frac{\pi }{4}\sin \theta )}$
$\Rightarrow f\left(\theta \right)=\frac{\cos \theta }{\sqrt{2}\cos (\frac{\pi }{4}-\theta )}=\frac{-\cos \theta }{\sqrt{2}\cos (\frac{5\pi }{4}-\theta )}$
$f\left(\alpha \right)f\left(\beta \right)=\left(\frac{-\cos \alpha }{\sqrt{2}\cos (\frac{5\pi }{4}-\alpha )}\right)\left(\frac{-\cos \beta }{\sqrt{2}\cos (\frac{5\pi }{4}-\beta )}\right)$
$\Rightarrow f\left(\alpha \right)f\left(\beta \right)=\frac{1}{2}\left(\frac{\cos \alpha }{\cos \beta }\right)\left(\frac{\cos \beta }{\cos \alpha }\right)\{\because \alpha +\beta =\frac{5\pi }{4}\}$
$\therefore f\left(\alpha \right)f\left(\beta \right)=\frac{1}{2}$
Ans: A