Proving identities

What a proof has to show

An identity says that two expressions are equal for every angle where both are defined. Checking $30^\circ$, $45^\circ$ and $60^\circ$ is a good start, but there are infinitely many angles. A proof shows the two sides are equal for all of them at once.

The rule is simple: work on one side only. Start with the left side, say. Change it, one true step at a time, until it becomes exactly the right side. Do not move terms across the equals sign, because that assumes what we are trying to prove.

Tools for the job

• The three identities: $\sin^2 A + \cos^2 A = 1$, $1 + \tan^2 A = \sec^2 A$ and $1 + \cot^2 A = \text{cosec}^2 A$.
• The definitions: $\tan A = \dfrac{\sin A}{\cos A}$, $\sec A = \dfrac{1}{\cos A}$, and so on.
• Ordinary algebra: common denominators, $(a + b)(a - b) = a^2 - b^2$, and factorising.

Worked example 1

Prove that $\sec A (1 - \sin A)(\sec A + \tan A) = 1$.

Left side. Write $\sec A$ and $\tan A$ with $\cos A$:

The top is $1 - \sin^2 A = \cos^2 A$. So the left side is $\dfrac{\cos^2 A}{\cos^2 A} = 1$, which is the right side.

Worked example 2

Prove that $\dfrac{\sin A}{1 + \cos A} = \dfrac{1 - \cos A}{\sin A}$.

Left side. Multiply the top and bottom by $1 - \cos A$:

Cancel one $\sin A$: the left side becomes $\dfrac{1 - \cos A}{\sin A}$, the right side.

Multiplying by $1 - \cos A$ is the key move here. It turns $1 + \cos A$ into $1 - \cos^2 A$, which an identity can simplify.

Worked example 3

Prove that $(\text{cosec}\,A - \cot A)(\text{cosec}\,A + \cot A) = 1$.

Left side. It is a difference of squares: $\text{cosec}^2 A - \cot^2 A$. By the third identity, this is $1$.

When a claim is false

Ananya claims that $(\sin A + \cos A)^2 = 1$ for every angle. Try $A = 45^\circ$: the left side is $\left(\dfrac{2}{\sqrt{2}}\right)^2 = 2$. One angle where it fails is enough. The claim is not an identity.

The correct expansion is $(\sin A + \cos A)^2 = \sin^2 A + 2\sin A\cos A + \cos^2 A = 1 + 2\sin A \cos A$.

Practise with new numbers

Try it in Python

A computer cannot prove an identity, but it can test a claim at many angles in a moment. If the claim fails anywhere, the program says where.

Common mistakes

• Working on both sides at once, then meeting in the middle with steps that only go one way.
• Cancelling terms, not factors. In $\dfrac{1 - \cos A}{\sin A}$, nothing cancels with the $1$.
• Squaring wrongly. $(\sin A + \cos A)^2$ is not $\sin^2 A + \cos^2 A$.
• Taking a few angles as a proof. Testing can show a claim is false. It cannot show it is always true.

Remember these

• Start with the more complicated side, and work on that side only.
• When stuck, write everything in $\sin A$ and $\cos A$.
• Look for $1 - \sin^2 A$, $1 - \cos^2 A$, $\sec^2 A - 1$ and $\text{cosec}^2 A - 1$: each is one term.
• One angle where a claim fails is enough to show it is not an identity.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. Prove that $\tan A \cos A = \sin A$.

2. Simplify $(1 + \tan^2 A)(1 - \sin^2 A)$.

3. Kavya tests a claim at $30^\circ$, $45^\circ$ and $60^\circ$, and it holds each time. Has she proved it?

4. Prove that $\dfrac{1}{1 - \sin A} + \dfrac{1}{1 + \sin A} = 2\sec^2 A$.

5. Show that $\sin A + \cos A = 1$ is not an identity.