Pre-Calculus 12 covers transformations of functions, exponential functions, geometric sequences and series, logarithms, polynomial functions, rational functions, and trigonometry - functions, equations and identities.
Ask any Surrey Pre-Calculus 12 class which of those was worst and the answer is almost unanimous: identities. Ask which unit they revised least and the answer is the same.
Why identities feel different
Everything else in Pre-Calculus 12 has a procedure. Solve for x. Find the inverse. Sketch the transformation. Identities have no procedure, and students who have got through two years of senior maths on reliable procedures suddenly have nothing to reach for.
What is actually being asked is narrow: show that two expressions are the same for all valid values. Not solve. Not simplify to a number. Show that one side can be rewritten as the other.
The rules we insist on
- Work on one side only. Pick the messier side and transform it until it matches the other. Doing algebra across the equals sign assumes the thing you are trying to prove, and it costs marks even when the final line is correct.
- Convert everything to sine and cosine first. Not always the shortest route. Almost always a route that works, which matters more under exam conditions than elegance does.
- Look for the Pythagorean identity in disguise. Any sum of squares, any 1 minus a square - that is where it hides.
- Common denominators are usually the move. When a side has two fractions, combining them is nearly always the step that unlocks it.
- State restrictions. Values where a denominator is zero are excluded, and marking guides ask for them.
Identities and equations are not the same task
This confusion costs real marks. A trigonometric equation has particular solutions and asks for them, usually within a stated interval, and usually with a general solution involving a period. An identity is true everywhere it is defined and asks for a proof.
A student who solves an identity gets nothing, and a student who proves an equation gets nothing. Reading the instruction word - prove, verify, solve - is worth doing deliberately before writing anything.
Practising it properly
Identities are one of the few topics where working through solutions is nearly useless. Reading a completed proof feels productive and teaches almost nothing, because the difficulty was choosing the first step and the finished proof hides that choice.
We practise them the other way round: student picks the first step out loud, explains why, and only then writes. Wrong first steps are useful, because the reason they fail is the thing worth learning. Twenty attempted proofs beats a hundred read ones.
Where it leads
Trigonometric identities reappear immediately in Calculus 12, where simplifying an expression before differentiating is often the whole difficulty of a question, and in Physics 12, where two-dimensional vector work runs on them. A student who leaves Pre-Calculus 12 shaky on identities meets them again within months.
That is the honest argument for dealing with them in Grade 12 rather than hoping they do not come back.
One worked identity, and the choice inside it
Prove that the expression sec x minus cos x equals sin x tan x.
Left side is messier, so start there. Convert to sine and cosine: one over cos x, minus cos x. Two terms, one of them a fraction, so combine over a common denominator. That gives one minus cos squared x, all over cos x.
Now the Pythagorean identity appears: one minus cos squared is sin squared. So the expression is sin squared x over cos x, which is sin x times sin x over cos x, which is sin x tan x.
Four steps. Every one of them came from a rule in the list above: pick the messier side, convert to sine and cosine, common denominator, look for the Pythagorean identity. There was no inspiration involved, and that is the point. Students think identities need cleverness. They need a short list of moves tried in order.
What to do when it stalls
Every student gets stuck partway through a proof. The productive response is a fixed sequence rather than staring:
- Is there a fraction that could be combined?
- Is there a squared term that could be a Pythagorean identity in disguise?
- Could a double-angle formula be applied backwards?
- Would starting from the other side be shorter?
Working the other side is legitimate, as long as one side is transformed at a time and the two are not mixed.
How much practice is enough
Fifteen attempted proofs, with the first step chosen out loud each time, is usually the point at which students stop finding them frightening. It is a small amount of work for a topic that carries a disproportionate share of the marks.
