Question

If ${\sin }^2(\frac{2}{1}\cdot \frac{3}{2}\cdot \frac{4}{3}.....\frac{x-1}{x-2})=1,0<x<100$, what is the value of $x$?

• A. $91$
• B. $80$
• C. $49$
• D. $46$

Answer 1

Answer: (A)

$91$

$\frac{2}{1}\cdot \frac{3}{2}\cdot \frac{4}{3}....\frac{x-1}{x-2}=x-1$ (Since the numerator and denominator of consecutive terms cancel each other out)

$\therefore {\sin }^2(x-1)=1$

$\Rightarrow \sin (x-1)=\pm 1$

$\Rightarrow x-1=\pm 90$

$\Rightarrow x=91$ or $x=-89$

Since $0<x<100$, $x=91$