If $\frac{cos\alpha}{cos\beta}=m$ and $\frac{cos\alpha}{sin\beta}=n$, then show that $( m^{2}+n^{2})cos^{2}\beta=n^{2}$.
Given: $\frac{cos\alpha}{cos\beta}=m$ and $\frac{cos\alpha}{sin\beta}=n$.
To do: To show that $( m^{2}+n^{2})cos^{2}\beta=n^{2}$.
Solution:
L.H.S.$=( m^{2}+n^{2})cos^{2}\beta$
$=( ( \frac{cos\alpha}{cos\beta})^{2}+( \frac{cos\alpha}{sin\beta})^{2})cos^{2}\beta$
$=( \frac{cos^{2}\alpha}{cos^{2}\beta}+\frac{cos^{2}\alpha}{sin^{2}\beta})cos^{2}\beta$
$=( \frac{cos^{2}\alpha.sin^{2}\beta+cos^{2}\alpha.cos^{2}\beta}{cos^{2}\beta.sin^{2}\beta})cos^{2}\beta$
$=cos^{2}\alpha( \frac{sin^{2}\beta+cos^{2}\beta}{cos^{2}\beta.sin^{2}\beta})cos^{2}\beta$
$=( \frac{cos^{2}\alpha}{sin^{2}\beta}.1.\frac{cos^{2}\beta}{cos^{2}\beta})$
$=\frac{cos^{2}\alpha}{sin^{2}\beta}$
$=n^{2}$
$=R.H.S.$
Hence, proved that $( m^{2}+n^{2})cos^{2}\beta=n^{2}$.
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