Question

sin5a-sin7a+sin8a-sin4a/cos4a+cos7a-cos5a-cos8a=cot6a

Answer 1

Given: The expression : sin 5a - sin 7a + sin 8a - sin 4a / cos 4a + cos 7a - cos 5a - cos 8a = cot 6a

To find: Prove LHS = RHS.

Solution:

• Now we have given the expression:

                sin 5a - sin 7a + sin 8a - sin 4a / cos 4a + cos 7a - cos 5a - cos 8a = cot 6a

• Consider LHS, we have:

                sin 5a - sin 7a + sin 8a - sin 4a / cos 4a + cos 7a - cos 5a - cos 8a

• Now we know the formulas:

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

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

• So using these formulas, we get:

                (sin 5a -sin 7a)+(sin 8a -sin 4a)/(cos 4a +cos 7a)-(cos 5a -cos 8a)

                2 sin (-2a/2) cos (12a/2) + 2 sin (2a/2) cos (12a/2) / -2 sin (12a/2) sin (-4a/2) + (-2)sin 12a/2 sin (2a/2)

• Simplifying it, we get:

                2sin 2a .cos 6a - 2sin a.cos 6a / 2sin 6a.sin 2a - 2sin 6a.sin a

• Taking 2 cos 6a and 2 sin 6a common from numerator and denominator, we get:

                2 cos 6a ( sin 2a - sin a ) / 2 sin 6a  ( sin 2a - sin a )

                cos 6a / sin 6a

                cot 6a ................RHS

                Hence proved.

Answer:

              So we proved that sin 5a - sin 7a + sin 8a - sin 4a / cos 4a + cos 7a - cos 5a - cos 8a = cot 6a .