Use Pythagorean, ratio, double angle and complementary identities to simplify expressions and prove equalities

What this dot point is asking

NESA wants you to know the standard trigonometric identities, choose the right one when simplifying or proving an equivalence, and use them to manipulate expressions involving $\sin$, $\cos$ and $\tan$, including double angle forms.

The skill being tested is not memory, it is selection: from a small toolbox of identities, picking the one that turns a messy expression into a clean one, or that transforms one side of a "prove that" into the other. Two habits separate full marks from partial: knowing that almost everything can be rewritten in terms of $\sin$ and $\cos$ alone (so when stuck, convert), and respecting the rule that a proof works down one side only, never juggling both sides at once. Get those two right and identity questions become mechanical.

The answer

The Pythagorean identity

For any angle $\theta$,

This is not an arbitrary rule to memorise; it is Pythagoras' theorem applied to the right triangle inside the unit circle. A point on the unit circle at angle $\theta$ has coordinates $(\cos\theta, \sin\theta)$, so the horizontal leg is $\cos\theta$, the vertical leg is $\sin\theta$, and the hypotenuse is the radius $1$. Pythagoras gives $\cos^2\theta + \sin^2\theta = 1^2$.

Two useful rearrangements (dividing by $\cos^2 \theta$ and $\sin^2 \theta$ respectively):

These let you swap freely between $\sin^2$ and $\cos^2$, between $\tan^2$ and $\sec^2$, and so on.

Ratio identities

Complementary angle identities

The complement of $\theta$ is $\frac{\pi}{2} - \theta$ (in degrees, $90^\circ - \theta$). Co-function pairs:

These come from triangle geometry: in a right triangle, $\sin$ of one acute angle equals $\cos$ of the other.

Supplementary, negative angle, and reflection identities

These follow from the symmetry of the unit circle.

Double angle identities

For $\sin$:

For $\cos$ (three equivalent forms, using the Pythagorean identity):

For $\tan$:

The choice of form for $\cos 2\theta$ depends on what you want to keep or eliminate.

Power reduction (useful when integrating)

From the double angle identities,

These convert squares of sine and cosine into linear expressions in $\cos 2\theta$, which is much easier to integrate.

Proof strategy

To prove an identity, start on one side (usually the more complicated) and transform it into the other using the identities above. Useful tactics:

• Convert everything to $\sin$ and $\cos$.
• Replace $\sin^2$ with $1 - \cos^2$ or vice versa.
• Apply a double angle identity when an angle is doubled or halved.
• Look for a common factor or a common denominator.

Do not start with the statement of the identity and manipulate both sides simultaneously. Take one side, work to the other, and conclude with "as required" or "QED".

A proof, stage by stage

Here is the 2022 HSC proof $\dfrac{1 - \cos 2\theta}{\sin 2\theta} = \tan\theta$ run as a chain of transformations. Read it top to bottom; each box is the line above with exactly one identity applied.

Stage 1, commit to the left-hand side

Choose the messier side, the fraction, and aim to reach $\tan\theta$. Never touch the right-hand side.

Stage 2, substitute the double angle identities

Replace $\cos 2\theta$ with the form $1 - 2\sin^2\theta$ (chosen because the leading $1$ will cancel) and $\sin 2\theta$ with $2\sin\theta\cos\theta$.

Stage 3, simplify the numerator

$1 - (1 - 2\sin^2\theta) = 2\sin^2\theta$, leaving $\dfrac{2\sin^2\theta}{2\sin\theta\cos\theta}$.

Stage 4, cancel to the right-hand side

Cancel the common $2\sin\theta$ to get $\dfrac{\sin\theta}{\cos\theta} = \tan\theta$, which is the right-hand side. As required.

How exam questions ask about identities

The verbs tell you what kind of answer earns the marks:

• "Simplify ..." Rewrite as a single, shorter expression. Look first for the Pythagorean identity hiding as $1 - \cos^2$, $1 - \sin^2$, $\sec^2 - 1$ or $\csc^2 - 1$.
• "Prove that ... " or "Show that LHS = RHS." Work down one side only to reach the other, and write a concluding line ("$= \text{RHS}$, as required"). Both-sides juggling scores zero for method.
• "Find the exact value of $\sin 2\theta$ given $\sin\theta = \dots$ and the quadrant." Get $\cos\theta$ from the Pythagorean identity, fix its sign from the quadrant, then apply the double angle formula.
• "Express ... in terms of $\cos 2\theta$ / a single trig function." A power-reduction or convert-to-one-function task: use $\sin^2\theta = \frac{1 - \cos 2\theta}{2}$ or replace $\tan$, $\sec$ etc. by $\sin$ and $\cos$.
• "Hence ... " after an identity part. Use the identity you just established as the substitution in the next part, typically to solve an equation or evaluate an integral.
• An expression mixing $\sin$, $\cos$, $\tan$, $\sec$. The default rescue move: rewrite everything in $\sin$ and $\cos$, put over a common denominator, and simplify.