If (cos(A-B)/cos(A+B))+ (cos(C+D)/cos(C-D)) = 0,prove that tan A tan B tan C tan D = -1

Question

If \(\frac{cos(A-B)}{cos(A+B)}\) + \(\frac{cos(C+D)}{cos(C-D)}\)= 0,

Prove that : tan A tan B tan C tan D = -1

Answer 1

Given,

(cos(A-B)/cos(A+B))+ (cos(C+D)/cos(C-D)) = 0

⇒ -1 = tan A tan B tan C tan D 

This proves that,

tan A tan B tan C tan D = -1