Let `f(theta)=cottheta/(1+cottheta)` and `alpha+beta=(5pi)/4` then the value ` f(alpha)f (beta)` is

Options

A. `(1)/(2)`
B. `-(1)/(2)`
C. 2
D. none of these.

Answer

A

Explanation

`f(beta)=f((5pi)/(4)-alpha)=(cot((5pi)/(4)-alpha))/(1+cot((5pi)/(4)-alpha))`
`=(1)/(1+(1-tanalpha)/(1+tan alpha))=(1+tan alpha)/(2)`
As `f(x)=(cotalpha)/(1+cotalpha)=(1)/(1+tan alpha)`, we have
`f(alpha)f(beta)=(1)/(2)`