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 answer

The Pythagorean identity

For any angle $\theta$,

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".

Worked examples

Simplify using Pythagoras

Simplify $\frac{1 - \cos^2 \theta}{\sin \theta}$.

$1 - \cos^2 \theta = \sin^2 \theta$, so the expression is $\frac{\sin^2 \theta}{\sin \theta} = \sin \theta$ (for $\sin \theta \neq 0$).

Prove an identity

Prove $\sec^2 \theta - 1 = \tan^2 \theta$.

LHS: $\sec^2 \theta - 1 = \frac{1}{\cos^2 \theta} - 1 = \frac{1 - \cos^2 \theta}{\cos^2 \theta} = \frac{\sin^2 \theta}{\cos^2 \theta} = \tan^2 \theta$ = RHS. As required.

Use a double angle identity

Express $\sin 2\theta \cos \theta$ in a form involving only single angles.

$\sin 2\theta \cos \theta = 2 \sin \theta \cos \theta \cdot \cos \theta = 2 \sin \theta \cos^2 \theta$.

If a question wanted everything in terms of $\sin \theta$, write $\cos^2 \theta = 1 - \sin^2 \theta$.

Find an exact value

Find $\cos 75^\circ$ given $\cos 30^\circ = \frac{\sqrt{3}}{2}$ and $\sin 30^\circ = \frac{1}{2}$.

$75^\circ = 90^\circ - 15^\circ$ does not directly use the identities here, but $\cos 75^\circ = \sin 15^\circ$ by the complementary identity. To get exact $\sin 15^\circ$, use sum/difference formulas (not in the dot point) or the half-angle identity from $\cos 30^\circ$:

$\sin^2 15^\circ = \frac{1 - \cos 30^\circ}{2} = \frac{1 - \sqrt{3}/2}{2} = \frac{2 - \sqrt{3}}{4}$, so $\sin 15^\circ = \frac{\sqrt{2 - \sqrt{3}}}{2}$ and $\cos 75^\circ$ has the same value.

Use power reduction

Rewrite $4 \sin^2 \theta$ in a form without squared trig.

$4 \sin^2 \theta = 4 \cdot \frac{1 - \cos 2\theta}{2} = 2 - 2 \cos 2\theta$.

Common traps

Treating $\sin 2\theta$ as $2 \sin \theta$. $\sin 2\theta = 2 \sin \theta \cos \theta$. The factor $\cos \theta$ is essential.

Dropping the sign in $\cos 2\theta$. All three forms ($\cos^2 - \sin^2$, $2 \cos^2 - 1$, $1 - 2 \sin^2$) are equivalent but easy to mis-write.

Forgetting the domain restriction. Identities like $\tan \theta = \frac{\sin \theta}{\cos \theta}$ require $\cos \theta \neq 0$. If a problem includes $\theta = \frac{\pi}{2}$, be careful.

Manipulating both sides simultaneously. When proving an identity, do not start from the conclusion and work both sides; that is not a valid proof. Pick one side and transform it.

Sign errors from the quadrant. When given a value of $\sin \theta$ and a quadrant, the sign of $\cos \theta$ (or $\tan \theta$) depends on the quadrant: positive in Q1; sine positive, cosine negative in Q2; both negative in Q3; cosine positive, sine negative in Q4.

In one sentence

The essential identities are the Pythagorean $\sin^2 \theta + \cos^2 \theta = 1$ (and its $\sec$/$\csc$ variants), the ratio identities for $\tan$, $\sec$, $\csc$, $\cot$, the complementary and reflection identities, and the double angle identities $\sin 2\theta = 2 \sin \theta \cos \theta$ and $\cos 2\theta = \cos^2 \theta - \sin^2 \theta$ in its three equivalent forms.