What is Wolfram doing?

Question

According to Wolfram Alpha,

$$\frac{T\cos^2(\pi m)}{\pi-2\pi m} + \frac{T\cos^2(\pi m)}{2\pi m + \pi}$$

has an alternate form

$$\frac{2T\cos^2(\pi m)}{\pi-4\pi m^2}$$

and I am not able to see the link. Can someone help?

Answer 1

Simple addition is required. Take the LCM of the denominator and proceed.

$$\require{cancel}\begin{align}\frac{T\cos^2(\pi m)}{\pi - 2\pi m} + \frac{T\cos^2(\pi m)}{\pi + 2\pi m} &= \frac{(\pi + 2\pi m)\left(T\cos^2(\pi m)\right) + (\pi - 2\pi m)\left(T\cos^2(\pi m)\right)}{(\pi - 2\pi m)(\pi + 2\pi m)} \\ &= \frac{\pi T\cos^2(\pi m) + \cancel{2\pi mT\cos^2(\pi m)} + \pi T\cos^2(\pi m) - \cancel{2\pi mT\cos^2(\pi m)}}{\pi^2 - \left(2\pi m\right)^2} \\ &= \frac{2\pi T\cos^2(\pi m)}{\pi^2 - 4\pi^2m^2} = \frac{2\cancel \pi T\cos^2(\pi m)}{\cancel \pi\left(\pi - 4\pi m^2\right)} = \frac{2T\cos^2(\pi m)}{\pi - 4\pi m^2}\end{align}$$