If $\dfrac{\text{cos A}}{\text{cos B}} = m \text{ and } \dfrac{\text{cos A}}{\text{sin B}}$= n,

show that:

(m² + n²) cos² B = n²

To prove:

(m² + n²) cos² B = n²

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

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

By formula,

⇒ sin² B + cos² B = 1

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

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

Hence, proved that (m² + n²) cos² B = n².