How to calculate `sin 4x`?

Explanation

`\cos 2 x \equiv 1-2 \sin ^{2} x`, or `\sin ^{2} x \equiv \frac{1}{2}(1-\cos 2 x)`


`\text { So } \sin ^{4} x \equiv(\sin ^{2} x)^{2} `




`\equiv(\frac{1}{2}(1-\cos 2 x))^{2} `


`\equiv \frac{1}{4}(1-2 \cos 2 x+\cos ^{2} 2 x)`

...