Introduction to Trigonometric Identities Revision Notes | GCSE AQA Higher Further Maths

1. Why Trigonometric Identities Matter

You already know $\sin\theta$, $\cos\theta$, and $\tan\theta$ as ratios in right-angled triangles. Trigonometric identities take that further: they are equations that hold true for every angle, not just a specific one. That makes them powerful tools for simplifying expressions and solving problems without a calculator.

There are two identities you need at GCSE. The first connects $\tan\theta$ to $\sin\theta$ and $\cos\theta$. The second is a Pythagorean relationship between $\sin\theta$ and $\cos\theta$. Both appear regularly in exam questions — sometimes hidden inside a longer problem.

2. The First Identity: $\tan\theta = \frac{\sin\theta}{\cos\theta}$

Think back to a right-angled triangle with angle $\theta$. The three sides are opposite (O), adjacent (A), and hypotenuse (H). You know $\sin\theta = \frac{O}{H}$, $\cos\theta = \frac{A}{H}$, and $\tan\theta = \frac{O}{A}$.

Now watch what happens when you divide $\sin\theta$ by $\cos\theta$: $\frac{\sin\theta}{\cos\theta} = \frac{O/H}{A/H} = \frac{O}{H} \times \frac{H}{A} = \frac{O}{A} = \tan\theta$. The hypotenuse cancels, and you land exactly on $\tan\theta$. This is where the identity comes from — it is not a rule to memorise blindly, it follows directly from the triangle ratios.

The condition $\cos\theta eq 0$ matters: if $\cos\theta = 0$ (which happens at $\theta = 90°$ and $\theta = 270°$), then $\tan\theta$ is undefined because you cannot divide by zero. Examiners occasionally test whether students recognise this restriction.

3. The Second Identity: $\sin^2\theta + \cos^2\theta = 1$

The second identity comes from Pythagoras' theorem applied to the unit circle — a circle of radius 1 centred at the origin. For any point on that circle at angle $\theta$, the coordinates are $(\cos\theta,\, \sin\theta)$. Pythagoras says $x^2 + y^2 = r^2$, so $\cos^2\theta + \sin^2\theta = 1^2 = 1$.

You do not need to derive this from the unit circle in the exam — you need to use it. The key skill is recognising when to rearrange it: $\sin^2\theta = 1 - \cos^2\theta$ and $\cos^2\theta = 1 - \sin^2\theta$ are both equally valid forms.

4. Using the Identities: Worked Examples

Knowing the identities is only half the job — the exam tests whether you can deploy them under pressure. The two most common tasks are: finding an unknown trig value given another, and simplifying or proving an expression. Here is how each one works.

Worked example: Finding $\cos\theta$ given $\sin\theta$

Given that $\sin\theta = \frac{3}{5}$ and $\theta$ is acute, find the exact value of $\cos\theta$.

1. $\sin^2\theta + \cos^2\theta = 1$ — Start with the identity — this is the tool for linking $\sin\theta$ and $\cos\theta$. [M1]
2. $\left(\frac{3}{5}\right)^2 + \cos^2\theta = 1$ — Replace $\sin\theta$ with $\frac{3}{5}$. Show the substitution explicitly for method marks.
3. $\frac{3}{5} \times \frac{3}{5} = \frac{9}{25}$, so $\frac{9}{25} + \cos^2\theta = 1$ — Multiply numerator by numerator and denominator by denominator. Do not skip this step on a non-calculator paper.
4. $\cos^2\theta = 1 - \frac{9}{25} = \frac{25}{25} - \frac{9}{25} = \frac{16}{25}$ — Subtract $\frac{9}{25}$ from both sides. Keep the fraction exact. [M1]
5. $\cos\theta = \sqrt{\frac{16}{25}} = \frac{4}{5}$ — Since $\theta$ is acute, $\cos\theta$ is positive. Take the positive root only. [A1]

Answer: $\cos\theta = \dfrac{4}{5}$

Worked example: Simplifying an expression using $\tan\theta = \frac{\sin\theta}{\cos\theta}$

Simplify $\frac{\sin\theta}{\tan\theta}$.

1. $\frac{\sin\theta}{\tan\theta} = \frac{\sin\theta}{\dfrac{\sin\theta}{\cos\theta}}$ — Substitute $\tan\theta = \frac{\sin\theta}{\cos\theta}$ into the denominator. [M1]
2. $= \sin\theta \times \frac{\cos\theta}{\sin\theta}$ — Dividing by $\frac{\sin\theta}{\cos\theta}$ is the same as multiplying by $\frac{\cos\theta}{\sin\theta}$.
3. $= \frac{\sin\theta \times \cos\theta}{\sin\theta} = \cos\theta$ — $\sin\theta$ appears in both numerator and denominator and cancels (provided $\sin\theta eq 0$). [A1]

Answer: $\cos\theta$

5. Proving Expressions Using Both Identities

Proof questions ask you to show that one expression equals another. The approach is always the same: start with the more complicated side, substitute using the identities, and simplify until you reach the target expression. Never work on both sides simultaneously — examiners expect a one-directional argument.

Worked example: Proving a trigonometric identity

Show that $\frac{\sin^2\theta}{1 - \cos\theta} = 1 + \cos\theta$.

1. LHS $= \frac{\sin^2\theta}{1 - \cos\theta}$ — Always start with the more complex side. State this clearly in your working.
2. $= \frac{1 - \cos^2\theta}{1 - \cos\theta}$ — From $\sin^2\theta + \cos^2\theta = 1$, we get $\sin^2\theta = 1 - \cos^2\theta$. Substitute this in. [M1]
3. $= \frac{(1 - \cos\theta)(1 + \cos\theta)}{1 - \cos\theta}$ — $1 - \cos^2\theta$ is a difference of two squares: $(1 - \cos\theta)(1 + \cos\theta)$. [M1]
4. $= 1 + \cos\theta$ = RHS — $(1 - \cos\theta)$ cancels from numerator and denominator (valid provided $\cos\theta eq 1$). This equals the right-hand side, so the identity is proved. [A1]

Answer: LHS $= 1 + \cos\theta =$ RHS. Identity proved.

6. Practice: Apply Both Identities

The worked examples above cover the main question types. Now try these problems yourself before checking the solutions — working through them independently is where the understanding sticks.

7. Exam Technique and Common Mistakes

These two identities appear in a range of question styles — from one-line simplifications worth one mark to multi-step proofs worth four or five marks. The method marks are generous, but only if your working is clear and flows logically from one line to the next.

For 'show that' and 'prove' questions, always work from one side only. Starting with both sides and meeting in the middle is not accepted as a valid proof at GCSE. Write 'LHS =' at the start of each line and end with '= RHS' to signal the proof is complete.

Both identities — $\tan\theta = \frac{\sin\theta}{\cos\theta}$ and $\sin^2\theta + \cos^2\theta = 1$ — are the core tools for the GCSE trigonometric identities specification point. Mastering when and how to substitute them is what separates a confident answer from a blank page.