A trigonometric identity is an equation that is true for every angle where both sides are defined.
For example:
\[
\sin^2\theta+\cos^2\theta=1
\]
This is an identity because it comes from the unit circle equation \(x^2+y^2=1\), with
\(x=\cos\theta\) and \(y=\sin\theta\).
1. Pythagorean identities
\[
\sin^2\theta+\cos^2\theta=1
\]
\[
1+\tan^2\theta=\sec^2\theta
\]
\[
1+\cot^2\theta=\csc^2\theta
\]
The second and third identities come from dividing the first identity by
\(\cos^2\theta\) or \(\sin^2\theta\).
2. Reciprocal identities
\[
\csc\theta=\frac{1}{\sin\theta}
\]
\[
\sec\theta=\frac{1}{\cos\theta}
\]
\[
\cot\theta=\frac{1}{\tan\theta}
\]
3. Quotient identities
\[
\tan\theta=\frac{\sin\theta}{\cos\theta}
\]
\[
\cot\theta=\frac{\cos\theta}{\sin\theta}
\]
4. Even and odd identities
Some trig functions have symmetry:
\[
\sin(-\theta)=-\sin\theta
\]
\[
\cos(-\theta)=\cos\theta
\]
\[
\tan(-\theta)=-\tan\theta
\]
5. Cofunction identities
Sine and cosine are related by complementary angles:
\[
\sin\left(\frac{\pi}{2}-\theta\right)=\cos\theta
\]
\[
\cos\left(\frac{\pi}{2}-\theta\right)=\sin\theta
\]
6. How to verify an identity
A common method is to transform one side until it matches the other side. Another check is to
subtract the two sides:
\[
\text{LHS}-\text{RHS}=0
\]
If the difference is always zero on the common domain, the identity is verified.
7. Common identity table
| Type |
Identity |
Important note |
| Pythagorean |
\(\sin^2\theta+\cos^2\theta=1\) |
Comes from the unit circle |
| Pythagorean |
\(1+\tan^2\theta=\sec^2\theta\) |
Valid where \(\cos\theta\ne0\) |
| Pythagorean |
\(1+\cot^2\theta=\csc^2\theta\) |
Valid where \(\sin\theta\ne0\) |
| Quotient |
\(\tan\theta=\dfrac{\sin\theta}{\cos\theta}\) |
Valid where \(\cos\theta\ne0\) |
| Quotient |
\(\cot\theta=\dfrac{\cos\theta}{\sin\theta}\) |
Valid where \(\sin\theta\ne0\) |
| Reciprocal |
\(\sec\theta=\dfrac{1}{\cos\theta}\) |
Valid where \(\cos\theta\ne0\) |
| Reciprocal |
\(\csc\theta=\dfrac{1}{\sin\theta}\) |
Valid where \(\sin\theta\ne0\) |
8. Common mistakes
- Do not verify an identity by testing only one angle.
- Remember that tangent and secant are undefined when \(\cos\theta=0\).
- Remember that cotangent and cosecant are undefined when \(\sin\theta=0\).
- Do not confuse \(\sin^2\theta\) with \(\sin(\theta^2)\).
- When proving an identity, work on one side at a time until it matches the other side.
