Please help me in this question.

Question

If $\frac{\sin^2\alpha}{\cos^2\beta} + \frac{\cos^2\beta}{\sin^2\beta} =1$ PROVE THAT $\frac{\sin^2\beta}{\cos^2\alpha} + \frac{\cos^2\beta}{\sin^2\alpha} = 1$ I tried a lot by using $\sin^2\alpha + \cos^2\beta =1$

Answer 1

HInt:

Assuming the condition to be

$$\dfrac{\sin^2\alpha}{\cos^2\beta}+\dfrac{\cos^2\alpha}{\sin^2\beta}=1$$

Use $\cos^2\alpha=1-\sin^2\alpha$ to get

$$1=\dfrac{\sin^2\alpha}{\cos^2\beta}+\dfrac{1-\sin^2\alpha}{\sin^2\beta}$$

$$\iff\csc^2\beta-1=\dfrac{\sin^2\alpha}{\sin^2\beta}-\dfrac{\sin^2\alpha}{\cos^2\beta}$$

$$\iff\dfrac{\cos^2\beta}{\sin^2\alpha}=\dfrac{\cos2\beta}{\cos^2\beta}$$

$$\iff\sin^2\alpha=\dfrac{\cos^4\beta}{\cos2\beta}\iff\cos^2\alpha=-\dfrac{\sin^4\beta}{\cos2\beta}$$

Now replace values of $\sin^2\alpha,\cos^2\alpha$ in

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