Double-angle and quadruple-angle identities for sine and cosine

Expert intro: These identities come up constantly in trigonometry and calculus. The cleanest way to build them is to start from the addition formulas, then simplify one step at a time so the pattern is easy to remember.

Detailed walkthrough

Use the standard angle-addition formulas.

1) Formula for $\sin 2A$

Set $B=A$ in the sine addition formula:

$\sin(A+B)=\sin A\cos B+\cos A\sin B$

So,

$\sin 2A=\sin(A+A)=\sin A\cos A+\cos A\sin A=2\sin A\cos A.$

2) Formula for $\cos 2A$

Start from the cosine addition formula:

$\cos(A+B)=\cos A\cos B-\sin A\sin B$

Thus,

$\cos 2A=\cos(A+A)=\cos^2A-\sin^2A.$

Using $\sin^2A+\cos^2A=1$, this can also be written as

$\cos 2A=2\cos^2A-1=1-2\sin^2A.$

3) Formula for $\sin 4A$

Treat $4A$ as $2(2A)$:

$\sin 4A=2\sin 2A\cos 2A.$

Now substitute the double-angle formulas:

$\sin 4A=2(2\sin A\cos A)(\cos^2A-\sin^2A).$

So,

$\sin 4A=4\sin A\cos A(\cos^2A-\sin^2A).$

You can also expand it if needed:

$\sin 4A=4\sin A\cos^3A-4\sin^3A\cos A.$

💡 Pitfall guide

A common slip is to write $\cos 2A=\cos^2A+\sin^2A$; that is just $1$, not the double-angle identity. Another one is mixing up $\sin 4A$ with $4\sin A\cos A$ — that expression is only for $\sin 2A$.

🔄 Real-world variant

If your teacher wants the answers in terms of only sine or only cosine, you can swap using $\sin^2A=1-\cos^2A$ or $\cos^2A=1-\sin^2A$. For example, $\cos 2A=1-2\sin^2A$ is often the most useful form when a problem already contains $\sin A$.

🔍 Related terms

double-angle identity, addition formulas, trigonometric identities