Angle sum and difference identities rewrite a trigonometric function of a combined angle,
such as \(\sin(75^\circ)\), using simpler angles such as \(45^\circ\) and \(30^\circ\).
1. Sine identities
\[
\sin(\alpha+\beta)=\sin\alpha\cos\beta+\cos\alpha\sin\beta
\]
\[
\sin(\alpha-\beta)=\sin\alpha\cos\beta-\cos\alpha\sin\beta
\]
2. Cosine identities
\[
\cos(\alpha+\beta)=\cos\alpha\cos\beta-\sin\alpha\sin\beta
\]
\[
\cos(\alpha-\beta)=\cos\alpha\cos\beta+\sin\alpha\sin\beta
\]
Notice the sign change: cosine uses the opposite sign inside the expanded expression.
3. Tangent identities
\[
\tan(\alpha+\beta)=\frac{\tan\alpha+\tan\beta}{1-\tan\alpha\tan\beta}
\]
\[
\tan(\alpha-\beta)=\frac{\tan\alpha-\tan\beta}{1+\tan\alpha\tan\beta}
\]
Tangent can be undefined if the denominator becomes zero, or if the combined angle has
cosine equal to zero.
4. Example: exact value of \(\sin75^\circ\)
Write \(75^\circ\) as \(45^\circ+30^\circ\):
\[
\sin75^\circ=\sin(45^\circ+30^\circ)
\]
\[
\sin75^\circ=\sin45^\circ\cos30^\circ+\cos45^\circ\sin30^\circ
\]
\[
\sin75^\circ=\frac{\sqrt2}{2}\cdot\frac{\sqrt3}{2}
+\frac{\sqrt2}{2}\cdot\frac12
\]
\[
\sin75^\circ=\frac{\sqrt6+\sqrt2}{4}
\]
5. Formula summary
| Function |
Sum identity |
Difference identity |
| Sine |
\(\sin(\alpha+\beta)=\sin\alpha\cos\beta+\cos\alpha\sin\beta\) |
\(\sin(\alpha-\beta)=\sin\alpha\cos\beta-\cos\alpha\sin\beta\) |
| Cosine |
\(\cos(\alpha+\beta)=\cos\alpha\cos\beta-\sin\alpha\sin\beta\) |
\(\cos(\alpha-\beta)=\cos\alpha\cos\beta+\sin\alpha\sin\beta\) |
| Tangent |
\(\tan(\alpha+\beta)=\dfrac{\tan\alpha+\tan\beta}{1-\tan\alpha\tan\beta}\) |
\(\tan(\alpha-\beta)=\dfrac{\tan\alpha-\tan\beta}{1+\tan\alpha\tan\beta}\) |
6. Common exact-angle decompositions
- \(75^\circ=45^\circ+30^\circ\)
- \(15^\circ=45^\circ-30^\circ\)
- \(105^\circ=60^\circ+45^\circ\)
- \(165^\circ=120^\circ+45^\circ\)
7. Common mistakes
- Do not write \(\sin(\alpha+\beta)=\sin\alpha+\sin\beta\). That is not true.
- For cosine, remember that \(\cos(\alpha+\beta)\) uses a minus sign.
- For tangent, check the denominator before simplifying.
- Keep angle units consistent: degrees with degrees, radians with radians.
- Use common angles like \(30^\circ\), \(45^\circ\), and \(60^\circ\) to get exact values.
