Question

sin 5x cos 2x + cos 6x sin 3x = sin 8x cosx

prove it..​

Answer 1

Given: sin 5x cos 2x + cos 6x sin 3x = sin 8x cosx

To find: Prove the above expression.

Solution:

• Now we have given the term sin 5x cos 2x + cos 6x sin 3x

• Multiplying and dividing it by 2, we get:

               1/2 (2 sin 5x cos 2x + 2 cos 6x sin 3x)

• Now we know the formula :

               2 sinx cosy = sin(x + y) + sin(x - y)

• So applying this, we get:

               1/2(sin(5x+2x) + sin(5x-2x) + sin(6x+3x) + sin(3x-6x))

• Simplifying this, we get:

               1/2(sin 7x + sin 3x + sin 9x + sin(-3x))

               1/2(sin 7x + sin 3x + sin 9x - sin 3x)

               1/2(sin 7x + sin 9x)

• Now we know the formula:

               sinx + siny = 2 sin x+y/2 cos x-y/2

• So applying this, we get:

               1/2(2 sin 9x + 7x / 2 cos 9x - 7x / 2)

               sin 16x/2 cos 2x/2

               sin 8x cos x

• Hence proved.

Answer:

             So in solution part we proved that sin 5x cos 2x + cos 6x sin 3x = sin 8x cosx