Why Students Get Trigonometry Proofs Wrong in A-Level Maths (And How to Fix It)

19 August 2026 · A-Level Maths

By Alex Fraley

Two distinct operational rules will eliminate almost every wasted line of algebra you write in A-Level trigonometry.

Whenever I sit down with an A-Level Maths student who feels stuck on trigonometry, the complaint is almost always the same: "I know all the formula booklet identities, but when I start manipulating the expression, I just go round in circles until I run out of page."

It is a completely understandable frustration. Unlike differentiation or integration, where standard rules (product rule, chain rule, integration by parts) tell you exactly what step to take next, trigonometric identities give you too much freedom. You could turn \(\tan(x)\) into \(\frac{\sin(x)}{\cos(x)}\), or replace \(\cos^2(x)\) with \(1 - \sin^2(x)\), or expand \(\cos(2x)\) in three different ways. Pick the wrong path, and your expression balloons into a messy algebraic monster.

To master trigonometry at this level, you do not need luck or flash-of-insight intuition. You need a systematic strategy based on whether you are solving an equation or proving an identity.

Solving Equations: The Two Golden Rules of Trigonometry

When an exam question asks you to solve an equation involving trigonometric terms (e.g. finding all values of \(\theta\) in the interval \(0 \le \theta < 2\pi\)), apply these two non-negotiable principles in order:

1. Harmonise the arguments first: Use identities so that every trigonometric function operates on the exact same angle (argument). If your equation mixes \(2\theta\) and \(\theta\), or \(\theta + \frac{\pi}{3}\) and \(\theta\), your first job is to get everything expressed in terms of a single argument (usually \(\theta\)).
2. Minimise the number of distinct trigonometric functions: Once all arguments match, rewrite the equation so that it uses as few different trig functions as possible—ideally reducing the entire expression to a single function (such as a quadratic in \(\sin(\theta)\) or a single equation in \(\tan(\theta)\)).

Let us see these two rules in action on a typical exam-style problem.

Example 1: Solve the equation \(\sin(2\theta) = 3\cos(\theta)\) for \(0^\circ \le \theta < 360^\circ\).

Step 1: Match the arguments. On the left-hand side, the argument is \(2\theta\); on the right, it is \(\theta\). We apply the double angle identity \(\sin(2\theta) \equiv 2\sin(\theta)\cos(\theta)\):

Now every trig function has the same argument, \(\theta\).

Step 2: Reduce functions without dividing by zero. A classic student mistake is dividing both sides by \(\cos(\theta)\), which loses solutions where \(\cos(\theta) = 0\). Instead, collect all terms on one side and factorise:

This splits neatly into two simple equations:

Since \(-1 \le \sin(\theta) \le 1\), \(\sin(\theta) = 1.5\) yields no real solutions. For \(\cos(\theta) = 0\) in the interval \(0^\circ \le \theta < 360^\circ\), we get:

By focusing purely on argument-matching and function-reduction, we avoided dead ends entirely.

Identity Proofs: How to "Force Out" the Target Form

Proving an identity—showing that the left-hand side (LHS) equals the right-hand side (RHS)—requires a different mindset. Many students start blindly expanding both sides, hoping the expressions magically meet in the middle.

Instead, look at the target expression on the RHS. Treat it as a blueprint. Your goal is to factorise out or subtract off components of the target answer piece by piece. By forcing the structure of the target into your working, you leave yourself with a much simpler, isolated sub-problem.

Let us work through a proof to see how this works in practice.

Example 2: Prove the identity:

Start with the more complex left-hand side:

Our target on the right-hand side is \(\cot(x) = \frac{\cos(x)}{\sin(x)}\). Notice two critical things about our target:

• The target has arguments of \(x\), not \(2x\).
• The denominator of the target has a factor of \(\sin(x)\), with no stray constants.

This tells us exactly which identities to pick. In the denominator, we must use \(\sin(2x) = 2\sin(x)\cos(x)\). In the numerator, we want to cancel the constant \(1\) so that we can eventually cancel factors with the denominator. Among the three forms for \(\cos(2x)\), only \(\cos(2x) = 2\cos^2(x) - 1\) eliminates that constant \(+1\):

Simplify the numerator:

Now cancel the common factor of \(2\cos(x)\) from both numerator and denominator:

Notice what happened: we did not guess. We chose the form of \(\cos(2x)\) specifically to subtract off the unwanted \(+1\) and force the single-term structure of the RHS.

A More Advanced Example: Forcing Out Common Factors

Consider a longer algebraic identity where students often get lost in expanded polynomials:

Example 3: Prove that \(\frac{\sin(3x) - \sin(x)}{\cos(3x) + \cos(x)} \equiv \tan(x)\).

If you attempt to expand \(\sin(3x) = \sin(2x + x)\) and \(\cos(3x) = \cos(2x + x)\), you can easily fill an entire page. But look at the target: \(\tan(x) = \frac{\sin(x)}{\cos(x)}\).

If we use the sum-to-product (factor) formulae directly on the LHS:

Applying these with \(P = 3x\) and \(Q = x\):

Now substitute them back into the fraction:

By recognising that we needed a solitary \(\sin(x)\) in the numerator and \(\cos(x)\) in the denominator, the sum-to-product formulae immediately forced out the matching factor \(2\cos(2x)\) for instant cancellation.

The Trigonometry Problem-Solving Checklist

Before writing down your next line of working, pause and ask yourself these three questions:

1. Are all arguments identical? If not, use double-angle or compound-angle identities to match them before doing anything else.
2. Can I reduce the number of trig functions? Use reciprocal identities, quotient identities (\(\tan = \frac{\sin}{\cos}\)), or Pythagorean identities (\(\sin^2 + \cos^2 = 1\)) to homogenise the expression.
3. What does the target form have that my current expression lacks? Factor out or cancel constants and coefficients systematically to reveal the target.

Trigonometric proofs are not creative writing exercises; they are structural transformations. Once you guide your algebraic steps using the target form rather than random trial-and-error, trigonometry becomes one of the most reliable scoring sections on your A-Level Maths papers.