Trigonometric Identities You Must Know for the Exam

Trig identities. I know — even saying the phrase makes half the room stare into space like they’ve left their brain in the corridor. And honestly, I get it: you sort of remember them until a question asks you to actually simplify something… and then the algebra turns into alphabet soup. The trick is realising that identities aren’t random — they behave like Lego bricks. Once you know which brick goes where, you stop guessing and start constructing.
And if you think of this topic as one of those A Level Maths concepts you must know, everything else in trigonometry becomes far less dramatic.

Let’s go slow — real teacher style, with pauses, scribble-energy, and no textbook voice.

🔙 Previous topic:

Earlier, we explored Binomial Expansion with negative and fractional indices, strengthening the algebra skills we now build on in Trigonometric Identities.

📌 Why examiners love identities (and why marks bleed here)

You’ll find trig identities in simplifications, proofs, double-angle questions, calculus, coordinate geometry, even mechanics. They’re everywhere.
The examiner isn’t just asking for an answer — they want to see whether you can choose the right identity instead of throwing every formula at the wall. It’s not roulette, and it shouldn’t feel like it.
Marks are won by using fewer steps, not more:

Pick the identity that shrinks the expression
Don’t jump between degrees & radians
Parentheses matter (like, really matter)
Identities are reversible — students forget this

Let’s build your toolbox cleanly before we twist anything complicated.

📏 Problem Setup — we need one expression to play with

We’re going to keep returning to this one:

For example, $\sin(2x) + \cos^2 x$

Not solving it yet — just parking it for later.

🧠 Step 1 — The three “root identities”

These are the foundations. Forget them and the building falls over.

Pythagorean backbone
For example, $\sin^2x + \cos^2x = 1$ .
From this, replace one with the other whenever needed — most simplifications are just this in disguise.
Tangent definition
Instead of memorising, think “tan is just sine over cosine”.
That one sentence does the job of multiple formulas.
Sec/cosec forms
They exist, but one is enough for A Level survival — sec-squared equals one plus tan-squared.
If you remember that single relationship, you’re covered.

No maths overload — just meaning.

📘 Step 2 — Double-angle identities (the examiner’s favourite toy)

Rather than writing four equations, we keep one example and explain the rest:

For example, $\sin(2x) = 2\sin x\cos x$ .
Cosine-double has multiple forms, but you choose the one that removes what you don’t want — either eliminate sine, or eliminate cosine.
Most errors here come from using the wrong variation at the wrong time.

This is exactly where A Level Maths revision mistakes to avoid matter — wrong identity choice → longer working → panic → lost marks.

📙 Step 3 — Compound-angle formulas without drowning in symbols

Again, one main formula → rest explained verbally:

For example, $\sin(A+B)=\sin A\cos B + \cos A\sin B$ .

Cosine looks similar but flips a sign. Tap that into memory gently:
sine keeps the sign, cosine flips one.
Tangent uses the same idea but wrapped in a fraction — no need to memorise it perfectly, as long as you know where its shape comes from.

A real teacher aside — please don’t try expanding $\sin(3x)$ as x + x + x. The walls weep when you do. Use double-angle + standard identities.

🧭 Step 4 — When stuck, convert everything to sine & cosine

This move looks too simple to be useful, yet it cracks open half the nasty questions.

For example, $\tan x\cot x = 1$ once rewritten using sine/cosine definitions.

When expression chaos begins → drop everything into sine & cosine → breathe → simplify.

🧩 Step 5 — Using identities to simplify properly

Return to our parked expression:

For example, $\sin(2x) + \cos^2 x$

Replace double-angle sine → becomes $2\sin x\cos x$
Replace cos² using Pythagorean → becomes $1 – \sin^2x$

So the whole expression becomes:

$2\sin x\cos x + 1 – \sin^2x$

We could go further, but we don’t need to — simplification stops when expression is smaller, not longer.

🔐 Step 6 — Memory hooks that actually stick

Not formal — but real-world classroom tips:

✔ “sin² + cos² = whole” — whole = 1
✔ “sin-double = two-sin-cos” — it even sounds right
✔ “cos-double = cos² minus sin²” — one neat phrase
✔ “cosine flips the sign” — saves half a page of panic

If identities feel like patterns instead of lists, recall speed rockets.

🧮 Worked Example — exam-format, not symbolic sludge

Simplify:

$\frac{1 – \cos(2x)}{\sin x}$

Use the cos-double rearrangement that removes cosine:

$1 – \cos(2x) = 2\sin^2 x$

So the expression becomes → $2\sin x$ .
Clean. Three steps. Human-realistic speed.

❗ Common Mistakes You Must Not Make

Mixing radians & degrees
Using wrong double-angle form
Expanding $\sin(3x)$ blindly
Forgetting parentheses
Simplifying too far instead of enough
Reflex-substituting everything → bigger, not smaller

Remember — identities work both directions. You can replace 1 − sin² with cos², or cos(2x) with its sine-only or cosine-only version.

🌍 Where this matters in real life

You use trig identities in wave modelling, signal processing, robotics, animation, acoustics, and 3D rotation mathematics.
Every time something oscillates — sine and cosine are running under the surface.
You’re not memorising — you’re learning the maths of movement.

🚀 Next Steps

If you want to stop guessing identities and instead feel which one belongs where, the A Level Maths Revision Course with guided practice builds identity fluency through worked examples, exam-traps, and proper teacher-style walkthroughs.

📏 Recap Table

Identity Type	Key Use
Pythagorean	swap sin² & cos² easily
Tangent	sin/cos unmasked
Double-angle	remove a function cleanly
Compound-angle	add/subtract angles
Convert-to-sine/cosine	panic emergency button

👤Author Bio – S Mahandru

After teaching A Level Maths for a decade, I can confidently say this: trig is rhythm, not memory. You need the right identity at the right moment, and when that clicks — the whole chapter calms down. Fast.

🧭 Next topic:

After the identities, Solving Trig Equations in Radians – Exam Style Walkthroughs is the natural next step, since those identities power every solution.

❓ FAQ

How do I choose which identity to use?

Most students panic and try everything — that’s the opposite of what works. Instead, look for what the examiner wants removed — if you see both sine and cosine, a Pythagorean swap might clean it; if you want only sine, replace cos². If you spot a double angle like 2x, pick the identity that shrinks it to single-x. You’re simplifying, not expanding the problem. With repetition, the choice becomes instinct — like shifting gears without thinking.

Should I memorise everything or rely on logic?

You don’t need to memorise every variant — you just need to understand how they grow from the root identities. If you remember one version of cos(2x), you can algebraically derive the other two in seconds. Compound-angle identities look big, but one memory pattern (sine keeps sign, cosine flips) unlocks them. Revision is easier when identities are relationships, not lists. You’re building recall through use, not flashcards alone.

What’s the best fix when I get stuck mid-question?

Pause. Redraw the expression mentally as sine/cosine — it strips off disguises. Nine times out of ten the solution appears once the clutter dies. Students think “more algebra” solves problems, but clarity does. Simplify the scene before charging forward — it saves marks and time.

Trigonometric Identities You Must Know for the Exam