Edexcel Pure Paper 2 2024 Question 8

Edexcel Pure Paper 2 2024 Question 8 – Trigonometry

❓ The Question

🧠 Before you start

At first glance, this looks quite full. There’s a lot written there, and the trig functions don’t exactly simplify themselves.

But if you step back for a second, it’s actually a very standard setup. One part is asking you to show something, and the next part uses it. That pattern comes up a lot.

The only real difficulty is resisting the urge to rush the first bit. If part (a) is slightly off, part (b) becomes awkward very quickly.

So it’s worth slowing down early, even if it feels like you’re overdoing it.

✏️ Working

Part (a)

We start with:

$\frac{1}{\csc\theta – 1} + \frac{1}{\csc\theta + 1}$

Step 1: Combine fractions

$= \frac{(\csc\theta + 1) + (\csc\theta – 1)}{(\csc\theta – 1)(\csc\theta + 1)}$

$= \frac{2\csc\theta}{\csc^2\theta – 1}$

Step 2: Use identity

$\csc^2\theta – 1 = \cot^2\theta$

So:

$= \frac{2\csc\theta}{\cot^2\theta}$

Step 3: Rewrite in sin/cos

$\csc\theta = \frac{1}{\sin\theta}, \quad \cot\theta = \frac{\cos\theta}{\sin\theta}$

So:

$\cot^2\theta = \frac{\cos^2\theta}{\sin^2\theta}$

Step 4: Simplify

\frac{2\csc\theta}{\cot^2\theta} = \frac{2\sin\theta}{\cos^2\theta}

Step 5: Final form

$= 2 \cdot \frac{\sin\theta}{\cos\theta} \cdot \frac{1}{\cos\theta}$

$= 2 \tan\theta \sec\theta$

Final answer (a):

$2 \tan\theta \sec\theta$

Part (b)

We are given:

$\frac{1}{\csc 2x – 1} + \frac{1}{\csc 2x + 1} = \cot 2x \sec 2x$

Step 1: Use result from part (a)

Replace LHS:

$2 \tan 2x \sec 2x = \cot 2x \sec 2x$

Step 2: Rearrange

$2 \tan 2x \sec 2x – \cot 2x \sec 2x = 0$

Factor:

$\sec 2x (2 \tan 2x – \cot 2x) = 0$

Step 3: Solve

$\sec 2x \neq 0$

So:

$2 \tan 2x – \cot 2x = 0$

Step 4: Rewrite

$2 \tan 2x = \cot 2x$

$\tan 2x = \frac{1}{2} \cot 2x$

Use:

$\cot 2x = \frac{1}{\tan 2x}$

So:

$2 \tan 2x = \frac{1}{\tan 2x}$

Step 5: Solve

$2 \tan^2 2x = 1$

$\tan^2 2x = \frac{1}{2}$

Step 6: Convert to sine/cosine

$\frac{\sin^2 2x}{\cos^2 2x} = \frac{1}{2}$

$2\sin^2 2x = \cos^2 2x$

Use:

$\cos^2 2x = 1 – \sin^2 2x$

So:

$2\sin^2 2x = 1 – \sin^2 2x$

$3\sin^2 2x = 1$

$\sin^2 2x = \frac{1}{3}$

Step 7: Solve trig

$\sin 2x = \frac{1}{\sqrt{3}}$

So:

$2x = \sin^{-1}\left(\frac{1}{\sqrt{3}}\right)$

Step 8: General solutions (0 < x < 90°)

$2x = 35.3^\circ \quad \text{or} \quad 180^\circ – 35.3^\circ = 144.7^\circ$

Step 9: Divide by 2

$x = 17.6^\circ \quad \text{or} \quad 72.4^\circ$

✅ Final answers (b):

$x = 17.6^\circ,; 72.4^\circ$

🎯 Where the Marks Are

Part (a) takes up most of the marks.

Not because it’s harder, but because there are more steps involved. Each bit of working counts.

Part (b) is shorter, but it still needs care — especially when listing all solutions.

⚠️ What Went Wrong

A lot of the issues came from trying to move too quickly through the proof.

Some answers switched between both sides of the identity, which tends to make things harder rather than easier. It’s usually better to just stick with one side and work it through.

There were also cases where everything wasn’t converted into sine and cosine early on. That made the expressions harder to manage, and the algebra became messy.

In part (b), the main problem wasn’t the solving itself — it was stopping after the first answer. That’s a really common habit, especially under time pressure.

💡 One Small Tip

If you’re proving something, commit to one side.

Trying to adjust both sides at once almost always leads to confusion.

🚀 If This Felt Difficult

If this didn’t feel smooth, that’s fairly normal.

Trigonometry tends to build up over time — identities, algebra, solving — and it’s when they all appear together that it feels heavier.

Working through more questions like this helps you get better at maths, mainly because you start to recognise what each part is asking for.

And if you prefer a clearer structure when revising, keeping things in an organised A level maths learning setup can make it easier to see how these topics connect.

🔗 Next Steps

← Question 7 Solution
→ Question 9 Solution

👨‍🏫Author Bio

S Mahandru is an A Level Maths teacher focused on helping students stay accurate under exam pressure. The aim is to keep methods clear and build confidence through consistent exam-style practice.

❓ Frequently Asked Questions

📌Do I always rewrite in sine and cosine?

Not always, but it usually makes identities easier to handle.

📌Is there a “best” way to prove identities?

Not exactly — but working from one side is generally safer.

📌How many solutions should I give?

As many as fall within the interval given.

📌Are these questions common?

Yes, especially combining identities with solving.

Edexcel Pure Paper 2 2024 Question 8