Question

Express the sum or difference as a product. sin 4x - sin 6x

Answer 1

Hello,

Since:

`sin (a)- sin(b)=cos( (a+b)/(2)) sin( (a-b)/(2) )\\ sin(4x)-sin(6x)=-(sin(6x)-sin(4x))=-2 cos(5x) sin(x)\\`

Answer 2

sin 4x - sin 6x = -2 cos 5x . sin x

Explanation

We use the following result to solve this expression,

`sin\,A-sin\,B=2\,cos\,((A+B)/(2))\,sin\,((A-B)/(2))`

and sin(-x) = -sin x

Consider,

sin 4x - sin 6x

`=2\,cos\,((4x+6x)/(2))\,sin\,((4x-6x)/(2))`

`=2\,cos\,((10x)/(2))\,sin\,((-2x)/(2))`

`=2\,cos\,5x\,sin\,(-x)`

`=2\,cos\,5x\,(-sin\,x)`

`=-2\,cos\,5x\,sin\,x`

Therefore, sin 4x - sin 6x = -2 cos 5x . sin x