Prove the following equation based on the picplease explain

Prove the following equation, based on the pic...please explain in detail.

Solution

Tan(A-B) = sin(A-B)/cos(A-B) Start with the given equation.


Tan(A-B) = sinA.cosB -cosA.sinA / cos(A-B) Expand sine using the sum difference identity


Tan(A-B) = sinA.cosB -cosA.sinB / cosA.cosB +sinA.sinB Expand cosine using the sum difference identity

now divide by cosA

Tan(A-B) = sinA/cosA. cosB -sinB / cosB + sinA/cosA.sinB

tan(A-B) = tanA.cosB - sinB / cosB + tanA.sinB

now divide with cosB

tan(A-B) = tanA.cosB/cosB -sinB/cosB / cosB/cosB + tanA.sinB/cosB

tan(A-B) = tanA - tanB/1+tanA.tanB

hence proved