Verify the identity

Question

Sin (A+B)/sin (A-B) = tan (A)+tan(B)/tan(A)-tan(B)

Answer 1

To verify the identity, we will use the following trigonometric identities:
1. Sum-to-product identities:
sin(A+B) = sinAcosB + cosAsinB
sin(A-B) = sinAcosB - cosAsinB

2. tanA = sinA/cosA

Starting with the left side of the equation:
sin(A+B) / sin(A-B)
= (sinAcosB + cosAsinB) / (sinAcosB - cosAsinB)
= [(sinAcosB + cosAsinB)/(sinAcosB - cosAsinB)] * [(1/cosA)/(1/cosA)] (Multiplying by cosA/cosA which is equivalent to 1)
= [(tanA + tanB) / (tanA - tanB)]

Therefore, the identity Sin (A+B)/sin (A-B) = tan (A)+tan(B)/tan(A)-tan(B) is verified.