Question

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

Answer 1

LHS :

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

$=(\frac{{\cos }^2\alpha }{{\cos }^2\beta }+\frac{co{s}^2\alpha }{si{n}^2\beta }){\cos }^2\beta$

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

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

$=\frac{{\cos }^2\alpha }{{\sin }^2\beta }=(\frac{\cos \alpha }{\sin \beta }{)}^2=n^2$ = RHS