Question

If $\frac{\cos \alpha }{\cos \beta }=m$ and $\frac{\cos \alpha }{\sin \beta }=n$, prove that $(m^2+n^2){\cos }^2\beta =n^2$.

Answer 1

$\frac{\cos \alpha }{\cos \beta }=m$ and $\frac{\cos \alpha }{\sin \beta }=n$

$\therefore \left(m^2+n^2\right){\cos }^2\beta$

$=\left(\frac{{\cos }^2\alpha }{{\cos }^2\beta }+\frac{{\cos }^2\alpha }{{\sin }^2\beta }\right).{\cos }^2\beta$

$=\left(\frac{{\cos }^2\alpha .{\sin }^2\beta +{\cos }^2\alpha .{\cos }^2\beta }{{\sin }^2\beta }\right)$

$={\cos }^2\alpha \left(\frac{{\sin }^2\beta +{\cos }^2\beta }{{\sin }^2\beta }\right)$

$=\frac{{\cos }^2\alpha }{{\sin }^2\beta }$

$=n^2$

Hence, proved.