What is a visual proof of the formulas $\cot(A-B)$ and $\cot(A+B)$?

Question

What is a visual proof of the formulas for $\cot(A-B)$ and $\cot(A+B)$?

I am looking for visual proof of trig identities. So, please explain mentioned trig identities in geometric way.

Answer 1

Start of proof: If you want a visual proof of, say, an identity like $\cot(A+B)=\cdots$, then the most natural thing to do would be to draw a right triangle where one angle is $A+B$, and from that angle draw a line segment to the opposite leg that makes an angle $A$. You now have two right triangles, one within the other, one with angle $A$, one with angle $A+B$, and you're ready to start calculating the ratios of sides to get an identity.