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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