Question

How do you express `sin(2x)+sin(4x)` in terms of `sin(x)` and `cos(x)` ?

Answer 1

In terms of `sin(x)` and `cos(x)` we find:

`sin(2x)+sin(4x) = 2 sin(x)cos(x)(1 + 2 cos^2(x) - 2 sin^2(x))`

Explanation:

Note that:

`sin(2x) = 2 sin(x)cos(x)`

`cos(2x) = cos^2(x) - sin^2(x)`

So:

`sin(4x) = sin(2(2x))`

`color(white)(sin(4x)) = 2 sin(2x)(cos(2x)`

`color(white)(sin(4x)) = 2(2sin(x)cos(x))(cos^2(x)-sin^2(x))`

`color(white)(sin(4x)) = 4sin(x)cos^3(x)-4sin^3(x)cos(x)`

So:

`sin(2x)+sin(4x) = 2 sin(x)cos(x) + 4sin(x)cos^3(x)-4sin^3(x)cos(x)`

`color(white)(sin(2x)+sin(4x)) = 2 sin(x)cos(x)(1 + 2 cos^2(x) - 2 sin^2(x))`