Question

sin^2a/ cos^2a + cos^2a/sin^2a = 1/sin^2a.cos^2a -2

Answer 1

$LHS=\frac{\sin^2 a}{\cos ^2 a} +\frac{\cos^2 a}{\sin ^2 a} \\ LHS=\frac{\sin^4 a+\cos ^4 a}{\sin ^2 a\cos ^2 a}$

Now use the formula,

$x^4+y^4=(x^2+y^2)^2-2x^2y^2$

$\sin^2 a+\cos ^2 a=1$

The equation becomes,

$LHS=\frac{(\sin^2 a+\cos ^2 a)^2-2\sin^2 a\cos ^2 a}{\sin ^2 a\cos ^2 a}\\ LHS=\frac{(1)^2-2\sin^2 a\cos ^2 a}{\sin ^2 a\cos ^2 a}\\ LHS=\frac{1}{\sin ^2 a\cos ^2 a}-2=RHS$

The proof is complete.