Given: The trigonometric term

            (cos A + cosB / sinA - sinB)^2015 +(sinA + sinB / cosA - cosB)^2015

To find: The value of the given expression?

SOLUTION:

• Now we have given the term as:

          (cos A + cosB / sinA - sinB)^2015 +(sinA + sinB / cosA - cosB)^2015

• Now we know the formulas :

          cos A + cosB = 2 cos(a+b/2) cos (a-b/2)

          sinA - sinB = 2 cos(a+b/2) sin(a-b/2)

          sinA + sinB = 2 sin(a+b/2) cos(a-b/2)

          cosA - cosB =  -2 sin(a+b/2) sin(a-b/2)

• Consider first term:

          (cos A + cosB / sinA - sinB)^2015

• So putting the values in the expression, we GET:

          (2 cos(a+b/2) cos (a-b/2))^2015 / (2 cos(a+b/2) sin(a-b/2))^2015

• Cancelling common terms, we get:

          (cos (a-b/2))^2015 / (sin(a-b/2))^2015

          (cot (a-b/2))^2015

• Consider second term:

          (sinA + sinB / cosA - cosB)^2015

• So putting the values in the expression, we get:

          (2 sin(a+b/2) cos(a-b/2))^2015 / (-2 sin(a+b/2) sin(a-b/2))^2015

• Cancelling common terms, we get:

          (cos (a-b/2))^2015 / (sin(a-b/2))^2015

          (cot (a-b/2))^2015

• Not TAKING them together, we get:

          (cot (a-b/2))^2015 - (cot (a-b/2))^2015

          0

Answer:

                So the answer of the given expression is 0.