i have to prove: tan(x-y)+tan(y-z)=(sec^2(y)(tan(x)-tan(z)) / (1+tan(x)tan(y))(1+tan(y)tan(z))

Question

I know that I have to expand the tan(x-y) and tan(y-z) with their subtraction formulas, but I get stuck when I have to add them together. How do I do that correctly? what do I do after that?

Thanks!

Answer 1

Transform both fractions using common demoninator:

(1 - tan x tan y)(1 - tan y tan z)

Add the two fractions, combine like terms.

Use 1 + tan² y = sec² y

It is just a lot of messy arithmetic.