Prove #cos(A) / sin(B) - sin(A) / cos(B) = (2 cos (A+B) )/ sin(2B)#

Multiply the first term by 1 in the form of #cos(B)/cos(B)#:

#cos(A) / sin(B)cos(B)/cos(B) - sin(A) / cos(B) = (2 cos (A+B) )/ sin(2B)#

Multiply the second term by 1 in the form of #sin(B)/sin(B)#:

#cos(A) / sin(B)cos(B)/cos(B) - sin(A) / cos(B)sin(B)/sin(B) = (2 cos (A+B) )/ sin(2B)#

Combine the two terms over the common denominator:

#(cos(A)cos(B) - sin(A)sin(B))/(sin(B)cos(B)) = (2 cos (A+B) )/ sin(2B)#

Use the identity #cos (A+B)=cos(A)cos(B) - sin(A)sin(B)#:

#cos (A+B)/(sin(B)cos(B)) = (2 cos (A+B) )/ sin(2B)#

Multiply by 1 in the form of #2/2#:

#(2cos (A+B))/(2sin(B)cos(B)) = (2 cos (A+B) )/ sin(2B)#

Use the identity #sin(2B) = 2sin(B)cos(B)#:

#(2cos (A+B))/sin(2B) = (2 cos (A+B) )/ sin(2B)# Q.E.D.