# What is the difference between the graph of f(x)=sin(4x) and that of f(x)=sin(5x)cosx-cos(5x)sinx?

Mar 7, 2017

They are same.

#### Explanation:

Graph of $f \left(x\right) = \sin \left(4 x\right)$ is as follows:
graph{sin(4x) [-5, 5, -2.46, 2.54]}

Graph of $g \left(x\right) = \sin \left(5 x\right) \cos x - \cos \left(5 x\right) \sin x$ is as follows:
graph{sin(5x)cosx-cos(5x)sinx [-5, 5, -2.46, 2.54]}

It is evident that the two are identical. The reason is that $\sin \left(4 x\right) = \sin \left(5 x\right) \cos x - \cos \left(5 x\right) \sin x$

This is as $\sin \left(A - B\right) = \sin A \cos B - \cos B \sin A$ and putting $A = 5 x$ and $B = x$ we get above result.