Question

Verify the following Identity-

Answer 1

we know that
sin(A + B) = sin A cos B + cos A sin B----> equation 1
sin(A − B) = sin A cos B − cos A sin B----> equation 2
so
tan A+tan B=[sin A/cos A]+[sin B/cos B]
=[(cos B*sin A)+(sin B*cos A)](cos A*cos B)
substitute equation 1
sin (A+B)/(cos A*cos B)

tan A-tan B=[sin A/cos A]-[sin B/cos B]
=[(cos B*sin A)-(sin B*cos A)](cos A*cos B)
substitute equation 2
sin (A-B)/(cos A*cos B)

[tan A+tan B]/[tan A-tan B]
=[sin (A+B)/(cos A*cos B)]/[sin (A-B)/(cos A*cos B)]
=[sin (A+B)/sin (A-B)]

hence
[tan A+tan B]/[tan A-tan B]=[sin (A+B)/sin (A-B)]