5.7 Trigonometric identities — Learn

Prerequisites: the definitions of sine, cosine and tangent; the exact values; the reciprocal functions.

By the end of this section you can: state the two basic trigonometric identities; derive the two reciprocal identities from them; prove a simple identity by working on one side.

1. The two basic identities

Two statements are true for every angle. They are identities, not equations, and the identity sign is often written $\equiv$ to make that clear.

This is Pythagoras' theorem on the unit circle: the point $(\cos\theta, \sin\theta)$ is always distance $1$ from the origin.

This is the definition of tangent as opposite over adjacent.

The notation $\sin^2\theta$ means $(\sin\theta)^2$. It is the square of the sine, not the sine of a square.

2. The reciprocal identities

Divide the first identity by $\cos^2\theta$:

which is

Divide the first identity by $\sin^2\theta$ instead:

which is

3. Proving an identity

To prove an identity, start with one side and change it, step by step, until it looks like the other side. Keep a chain of equals signs down the page. Do not move terms from one side to the other, because that assumes the result you are trying to prove.

A useful order of attack:

1. Write everything in terms of sine and cosine.
2. Combine fractions over a common denominator.
3. Use $\sin^2\theta + \cos^2\theta = 1$ to simplify.
4. Factorise if a common factor appears.

Example 1. Prove that $\dfrac{1 - \cos^2\theta}{\sin\theta} \equiv \sin\theta$ for $\sin\theta \neq 0$.

Start with the left-hand side and replace $1 - \cos^2\theta$ using $\sin^2\theta + \cos^2\theta = 1$:

The left-hand side becomes the right-hand side, so the identity is proved.

Example 2. Prove that $\tan\theta\cos\theta \equiv \sin\theta$ for $\cos\theta \neq 0$.

Replace tangent with sine over cosine:

Example 3. Prove that $(\sin\theta + \cos\theta)^2 \equiv 1 + 2\sin\theta\cos\theta$.

Expand the square:

Group the two squares and use the first identity:

Watch out. Two mistakes are common when proving identities.

Working on both sides at once. Moving terms across the equals sign assumes the identity is already true. Start from one side, usually the more complicated one, and keep the other side untouched until the last line.

Cross-multiplying to "simplify". Multiplying both sides by a factor that could be zero is not allowed. An identity such as $\dfrac{\sin\theta}{\cos\theta} \equiv \tan\theta$ is defined only where $\cos\theta \neq 0$. Re-check: finish with the left-hand side exactly equal to the right-hand side, and note any value of $\theta$ where a denominator vanishes.

You should now be able to

• state $\sin^2\theta + \cos^2\theta \equiv 1$ and $\tan\theta \equiv \dfrac{\sin\theta}{\cos\theta}$;
• derive $\sec^2\theta \equiv 1 + \tan^2\theta$ and $\operatorname{cosec}^2\theta \equiv 1 + \cot^2\theta$;
• prove a simple identity by changing one side only;
• convert a mixed expression into sines and cosines as a first move;
• note where a denominator vanishes.