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Question 160
If $$\frac{\cos\alpha}{\sin\beta}=n$$ and $$\frac{\cos\alpha}{\cos\beta}=m$$ then the value of $$\cos^{2} \beta$$ is
Solution
$$\frac{\cos\alpha}{\sin\beta}=n$$
=> $$cos\alpha = nsin\beta$$
and $$\frac{\cos\alpha}{\cos\beta}=m$$
=> $$cos\alpha = mcos\beta$$
Comparing above equations, we get :
=> $$nsin\beta = mcos\beta$$
Squaring both sides :
=> $$n^2sin^2\beta = m^2cos^2\beta$$
=> $$n^2(1-cos^2\beta) = m^2cos^2\beta$$
=> $$n^2 = (n^2+m^2)cos^2\beta$$
=> $$cos^2\beta = \frac{n^2}{m^2+n^2}$$
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