$\frac{1}{\sin^2(x)}+\frac{1}{\cos^2(x)} = \cos^2(x)+\sin^2(x) ?$ This is a simple question.  Since $\cos(\theta)^2 + \sin(\theta)^2 = 1$
Can I take the inverse of this $\frac{1}{\cos^2(\theta)}+\frac{1}{\sin^2(\theta)} = \frac{1}{1}$?
Finally getting $\frac{1}{\cos^2(\theta)}+\frac{1}{\sin^2(\theta)} = \cos(\theta)^2 + \sin(\theta)^2$
If this is incorrect thinking can someone please put me on the right path?
Thanks

It is not true that the reciprocal of $a+b$ is the reciprocal of $a$ plus the reciprocal of $b$.  The reciprocal of $\sin^2\theta+\cos^2\theta$ is not the reciprocal of $\sin^2\theta$ plus the reciprocal of $\cos^2\theta$.
