Prove identity (tan A + tan B) / (tan A - tan B) = (sin (A+B)) / (sin (A-B))

Question

I have to prove the identity :

 

(tan A + tan B) / (tan A - tan B) ≡ (sin (A+B)) / (sin (A-B)) 

 

 

Please help me with a detailed solution and the detailed steps, that would be great! 

Answer 1

When confused you can always start by converting tan to sin/cos.

 

(sinA/cosA + sinB/cosB)  /  (sinA/cosA - sinB/cosB)

 

 

multiply everything by cosAcosB / cosAcosB

 

(sinA cosB + sinB cosA)  /  (sinA cosB - sinB cosA)

 

 

which is recognizably the formula

 

sin(A+B)/sin(A-B)