Question

If $f\left(\frac{y}{y-1}\right)=\frac{1}{y}$ for all $y\ne 0,1$ 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. ${\mathrm{cosec}}^{2}x$
• D. ${\sec }^{2}x$

Answer 1

Answer: (B)

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