Question

If $f \left(\frac{ y }{ y -1}\right)=\frac{1}{ y }$ for all $y \neq 0,1$ and $0< x <\frac{\pi}{2}$, then an expression for $f \left(\csc ^2 x \right)$ would be

• A. $\sin ^2 x$
• B. $\cos ^2 x$
• C. $\operatorname{cosec}^2 x$
• D. $\sec ^2 x$

Answer 1

Answer: (B)

We have $f\left(\frac{y}{y-1}\right)=\frac{1}{y}$
Let $\frac{y}{y-1}=t \Rightarrow t y-t=y $
$ \Rightarrow y(t-1)=t \Rightarrow y=\frac{t}{t-1}$ $f(t)=\frac{t-1}{t}$
$f\left(\operatorname{cosec}^2 x\right)=\frac{\cot ^2 x}{\operatorname{cosec}^2 x}=\cos ^2 x$