how do you prove that

tanx+coty / tanxcoty = tany+cotx

\frac{tan(x)+cot(y)}{tan(x)cot(y)}

This only requires about one step.

Expand it out and look at the cancellations.

\frac{tan(x)}{tan(x)cot(y)}+\frac{cot(y)}{tan(x)co t(y)}

See it?

Thank you! I get it now!

how do I prove that sin20=x/8?

If I read your question correctly,

sin(20)=\frac {x}{8} ?

That looks more like an equation than a proof.

x=8 sin(20) \approx 2.736