2.2 Trigonometric Identities — Complete Knowledge Points
Chapter 2 · 2.2 | Trigonometric Identities
Contents
What is a Trig Identity? Pythagorean Identities
Reciprocal & Quotient Identities
Sum and Difference Formulas Double Angle Formulas Half Angle Formulas Co-function Identities Even and Odd Identities How to Prove a Trig Identity
A trigonometric identity is an equation involving
trig functions that is
true for all valid values of $\theta$ . It is
different from a trig equation , which is only true
for specific values.
Example
$\sin^2\theta + \cos^2\theta = 1$
Example
$\sin\theta = \dfrac{1}{2}$
True for
Only specific $\theta$
These are the most fundamental identities in
trigonometry. They all come from the unit circle equation $x^2 + y^2 =
1$. Since $x = \cos\theta$ and $y = \sin\theta$:
$$\sin^2\theta + \cos^2\theta = 1$$
This is the master identity . The other two are
derived from it:
Divide everything by $\cos^2\theta$
$$\frac{\sin^2\theta}{\cos^2\theta} +
\frac{\cos^2\theta}{\cos^2\theta} = \frac{1}{\cos^2\theta}$$
$$\boxed{\tan^2\theta + 1 = \sec^2\theta}$$
Divide everything by $\sin^2\theta$
$$\frac{\sin^2\theta}{\sin^2\theta} +
\frac{\cos^2\theta}{\sin^2\theta} = \frac{1}{\sin^2\theta}$$
$$\boxed{1 + \cot^2\theta = \csc^2\theta}$$
💡
You only need to memorize the first one — the other
two are derived by simple division, so you can always re-derive them
on the spot.
Reciprocal Identities
$$\csc\theta = \frac{1}{\sin\theta} \qquad \sec\theta =
\frac{1}{\cos\theta} \qquad \cot\theta = \frac{1}{\tan\theta}$$
Quotient Identities
$$\tan\theta = \frac{\sin\theta}{\cos\theta} \qquad \cot\theta =
\frac{\cos\theta}{\sin\theta}$$
These let you compute trig values of
sums or differences of angles :
$$\sin(\alpha \pm \beta) = \sin\alpha\cos\beta \pm
\cos\alpha\sin\beta$$
$$\cos(\alpha \pm \beta) = \cos\alpha\cos\beta \mp
\sin\alpha\sin\beta$$
$$\tan(\alpha \pm \beta) = \frac{\tan\alpha \pm \tan\beta}{1 \mp
\tan\alpha\tan\beta}$$
⚠️
Notice the $\mp$ in the cosine formula — the sign
flips . This is the most common mistake.
Want to see where these formulas come from?
Deriving the Sum and Difference Formulas
Practical Use — Computing $\sin 15°$
Example
$$\sin 15° = \sin(45° - 30°)$$ $$= \sin 45°\cos 30° - \cos 45°\sin
30°$$ $$= \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} -
\frac{\sqrt{2}}{2} \cdot \frac{1}{2}$$ $$= \frac{\sqrt{6}}{4} -
\frac{\sqrt{2}}{4} = \frac{\sqrt{6} - \sqrt{2}}{4}$$
These are a special case of the sum formulas where
$\alpha = \beta = \theta$:
For Sine
Set $\alpha = \beta = \theta$ in $\sin(\alpha + \beta)$
$$\sin(\theta + \theta) = \sin\theta\cos\theta +
\cos\theta\sin\theta$$ $$\boxed{\sin 2\theta =
2\sin\theta\cos\theta}$$
For Cosine
Set $\alpha = \beta = \theta$ in $\cos(\alpha + \beta)$ — three
equivalent forms
$$\boxed{\cos 2\theta = \cos^2\theta - \sin^2\theta = 2\cos^2\theta -
1 = 1 - 2\sin^2\theta}$$
For Tangent
Set $\alpha = \beta = \theta$ in $\tan(\alpha + \beta)$
$$\boxed{\tan 2\theta = \frac{2\tan\theta}{1 - \tan^2\theta}}$$
💡
The three forms of $\cos 2\theta$ are all equally valid. In
problems, you pick whichever form is most convenient.
These are derived directly from the double angle formulas for cosine.
From $\cos 2\theta = 1 - 2\sin^2\theta$, solve for $\sin^2\theta$,
then replace $\theta$ with $\frac{\theta}{2}$
$$\boxed{\sin\frac{\theta}{2} = \pm\sqrt{\frac{1 - \cos\theta}{2}}}$$
From $\cos 2\theta = 2\cos^2\theta - 1$, solve for $\cos^2\theta$,
then replace $\theta$ with $\frac{\theta}{2}$
$$\boxed{\cos\frac{\theta}{2} = \pm\sqrt{\frac{1 + \cos\theta}{2}}}$$
⚠️
The $\pm$ sign depends on which
quadrant $\dfrac{\theta}{2}$ lies in. Always
determine the sign from context.
These describe the relationship between
complementary angles ($\alpha + \beta = 90°$):
$$\sin\theta = \cos\left(\frac{\pi}{2} - \theta\right) \qquad
\cos\theta = \sin\left(\frac{\pi}{2} - \theta\right)$$ $$\tan\theta =
\cot\left(\frac{\pi}{2} - \theta\right) \qquad \cot\theta =
\tan\left(\frac{\pi}{2} - \theta\right)$$
💡
This is why $\sin 30° = \cos 60°$ and $\sin 60° = \cos 30°$ — they
are complementary pairs.
These describe what happens when you
negate the angle :
$$\sin(-\theta) = -\sin\theta \quad \text{(odd function)}$$
$$\cos(-\theta) = \cos\theta \quad \text{(even function)}$$
$$\tan(-\theta) = -\tan\theta \quad \text{(odd function)}$$
💡
This connects directly back to
even and odd functions from Section 1.2 — cosine is
even, sine and tangent are odd.
When asked to prove an identity, follow this standard approach:
01
Pick one side — usually the more complicated
side.
02
Rewrite it step by step using known identities.
03
Keep going until it matches the other side .
⚠️
You must never move terms across the equals sign —
that assumes what you are trying to prove. Always transform one side
independently.
Example — Prove $\tan\theta\cos\theta = \sin\theta$
Start from the left side
$$\tan\theta\cos\theta = \frac{\sin\theta}{\cos\theta} \cdot
\cos\theta = \sin\theta \checkmark$$
✓ Distinguish between a trig identity and
a trig equation
✓ Derive all three Pythagorean identities
from $\sin^2\theta + \cos^2\theta = 1$
✓ Apply reciprocal and quotient
identities correctly
✓ Use sum and difference formulas to
compute exact trig values
✓ Derive and apply all double angle
formulas
✓ Derive and apply half angle formulas,
with correct sign from context
✓ Use co-function and even/odd identities
✓ Prove a trig identity by transforming
one side only
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