Simplify
Simplify sin2x + cos2x + tan2x.
sin2x + cos2x = 1, so the expression becomes 1 + tan2x = sec2x.
Trig Identities
The identities that appear most often in exams and calculus, grouped by how they are used.
Pythagorean identities
| Identity | Equivalent forms |
|---|---|
sin2θ + cos2θ = 1 | sin2θ = 1 − cos2θ |
1 + tan2θ = sec2θ | tan2θ = sec2θ − 1 |
1 + cot2θ = csc2θ | cot2θ = csc2θ − 1 |
Reciprocal and quotient
| Identity | Identity |
|---|---|
csc θ = 1 / sin θ | sec θ = 1 / cos θ |
cot θ = 1 / tan θ | tan θ = sin θ / cos θ |
cot θ = cos θ / sin θ | sin2θ = (1 − cos 2θ) / 2 |
Sum and difference
| Identity | Formula |
|---|---|
sin(A ± B) | sin A cos B ± cos A sin B |
cos(A ± B) | cos A cos B ∓ sin A sin B |
tan(A ± B) | (tan A ± tan B) / (1 ∓ tan A tan B) |
Double and half angle
| Identity | Formula |
|---|---|
sin 2θ | 2 sin θ cos θ |
cos 2θ | cos2θ − sin2θ = 2cos2θ − 1 = 1 − 2sin2θ |
tan 2θ | 2 tan θ / (1 − tan2θ) |
sin(θ/2) | ±√((1 − cos θ)/2) |
cos(θ/2) | ±√((1 + cos θ)/2) |
Exact values
| Angle | sin | cos | tan |
|---|---|---|---|
| 0° | 0 | 1 | 0 |
| 30° | 1/2 | √3/2 | 1/√3 |
| 45° | √2/2 | √2/2 | 1 |
| 60° | √3/2 | 1/2 | √3 |
| 90° | 1 | 0 | undefined |
Worked examples
Double angle
Given sin θ = 3/5 and θ is acute, find cos 2θ.
cos 2θ = 1 − 2sin2θ = 1 − 2(9/25) = 7/25.
FAQ
Which cos 2θ form should I use?
Choose the form that matches what you already know: use 1 − 2sin²θ when you know sin θ, and 2cos²θ − 1 when you know cos θ.
How do I remember the Pythagorean identities?
Start with sin² + cos² = 1, divide by cos² to get 1 + tan² = sec², and divide by sin² to get 1 + cot² = csc².
Last updated November 1, 2026. Formulas are checked against standard references. Read the editorial policy.