If $\dfrac{\cos A}{\cos B}$ = m and $\dfrac{\cos A}{\sin B}$ = n, prove that : (m² + n²)cos²B = n²

To prove:

(m² + n²)cos²B = n²

Substituting value of m and n in L.H.S. of the above equation :

$\Rightarrow \Big[\Big(\dfrac{\cos A}{\cos B}\Big)^2 + \Big(\dfrac{\cos A}{\sin B}\Big)^2 \Big] \cos^2B \\[1em] \Rightarrow \Big[\dfrac{\cos^2 A}{\cos^2 B} + \dfrac{\cos^2 A}{\sin^2 B} \Big] \cos^2B \\[1em] \Rightarrow \Big[\dfrac{\cos^2 A \sin^2 B + \cos^2 A\cos^2 B}{\cos^2 B\sin^2 B} \Big] \cos^2 B \\[1em] \Rightarrow \dfrac{\cos^2 A (\sin^2 B + \cos^2 B)}{\sin^2 B} \\[1em] \Rightarrow \dfrac{\cos^2 A}{\sin^2 B} \\[1em] \Rightarrow \Big(\dfrac{\cos A}{\sin B}\Big)^2 \\[1em] \Rightarrow n^2$

Since, L.H.S. = R.H.S.

Hence, proved that (m² + n²)cos²B = n².