Question

If $\frac{\cos \alpha }{\sin \beta }=n$ and $\frac{\cos \alpha }{\cos \beta }=m$, then the value of ${\cos }^2\beta$ is

• A. $\frac{1}{m^2+n^2}$
• B. $\frac{m^2}{m^2+n^2}$
• C. $\frac{n^2}{m^2+n^2}$
• D. $0$

Answer 1

The correct option is C $\frac{n^2}{m^2+n^2}$

$\frac{\cos \alpha }{\sin \beta }=n$ and $\frac{\cos \alpha }{\cos \beta }=m$
$\Rightarrow \cos \alpha =n\sin \beta$ and $\cos \alpha =m\cos \beta$
$\therefore n^2{\sin }^2\beta =m^2{\cos }^2\beta$
$n^2(1-{\cos }^2\beta )=m^2{\cos }^2\beta$
$n^2-n^2{\cos }^2\beta =m^2{\cos }^2\beta$
$m^2{\cos }^2\beta +n^2{\cos }^2\beta =n^2$
${\cos }^2\beta (m^2+n^2)=n^2$
${\cos }^2\beta =\frac{n^2}{m^2+n^2}$.