Why is $\frac{20+16\cos{\alpha}}{25+20\cos{\alpha}}=\frac45$

Question

I am doing a question about conic sections and in order to prove a given theorem it is required to prove the following: $$\frac{20+16\cos{\alpha}}{25+20\cos{\alpha}}=\frac{20-16\cos{\alpha}}{25-20\cos{\alpha}}=\frac45$$ For all $\alpha\in\mathbb{R}$. I do not understand how this simplification is made as the supplied worked answer simply writes $\frac{20+16\cos{\alpha}}{25+20\cos{\alpha}}=\frac45$ with no intermediate working.

Answer 1

They've cancelled a factor of $5+4\cos\alpha$. If you struggle to spot such factors, try factorising the numerator and denominator.