Trigonometric Identities and Formulae

IB AA SL: core and optional content

Core: tan A=sin A/cos A and sin²A+cos²A=1, with their applications. Optional: cot, sec and cosec identities and examples.

An identity holds for every angle where both sides are defined.

Quotient identities

$\tan A=\dfrac{\sin A}{\cos A}\qquad\cot A=\dfrac{\cos A}{\sin A}$

Require nonzero denominators. The form $\cot A=1/\tan A$ also requires $\tan A$ to exist and be nonzero.

Pythagorean identities

$\sin^2A+\cos^2A=1$

$1+\tan^2A=\sec^2A\qquad1+\cot^2A=\operatorname{cosec}^2A$

The last two follow by dividing by cosine squared or sine squared respectively. Use the same angle throughout.

Correct use	Not an identity
$\sin^2 3\theta+\cos^2 3\theta=1$
$\sec^2\dfrac{\theta}{2}=1+\tan^2\dfrac{\theta}{2}$

Proving an identity

Start with one side, express reciprocal ratios in sine and cosine, then combine fractions or factorise until the other side is obtained.

Move	Working
Express in $\sin$ and $\cos$
Pythagorean swap after factorising
Combine into one fraction

Do not assume the identity you are proving.

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Correct use	Not an identity
\(\sin^2 3\theta+\cos^2 3\theta=1\)
\(\sec^2\dfrac{\theta}{2}=1+\tan^2\dfrac{\theta}{2}\)

Move	Working
Express in \(\sin\) and \(\cos\)
Pythagorean swap after factorising
Combine into one fraction