Question

If $\frac{\cos A}{\cos B}=m$ and $\frac{\cos A}{\sin B}=n.$Prove:- $(m^2+n^2){\cos }^{2}B=n$

Answer 1

$m^2=\frac{{\cos }^{2}A}{{\cos }^{2}B};n^2=\frac{{\cos }^{2}A}{{\sin }^{2}B}...\left(1\right)$

$\therefore m^2+n^2=\frac{{\cos }^{2}A}{{\sin }^{2}B{\cos }^{2}B}$

$(m^2+n^2){\cos }^{2}B=\frac{{\cos }^{2}A}{{\sin }^{2}B}...\left(2\right)$

from (1) and (2)

$(m^2+n^2){\cos }^{2}B=n^2$