Given,

cos (A + B) sin (C – D) = cos (A – B) sin (C + D)

Adding 1 both sides:

Now,

Subtracting 1 both sides:

Dividing equation (i) by equation (ii):

⇒ -tan D = tan A tan B tan C

⇒ tan D = -tan A tan B tan C

⇒ tan A tan B tan C + tan D = 0

Hence Proved