RevisionDojo

Whether you’re preparing for AP Calculus, Precalculus, or any math course involving trigonometry, knowing the core trigonometric identities is essential. They help you simplify expressions, prove equalities, and solve equations — and they appear on AP exams, SAT Math, and in college-level courses.

In this guide, you’ll find every major trig identity, along with tips for remembering them and using them effectively on exams.

1. Reciprocal Identities

These express trig functions as reciprocals of each other:

sin⁡θ=1csc⁡θ\sin\theta = \frac{1}{\csc\theta}
csc⁡θ=1sin⁡θ\csc\theta = \frac{1}{\sin\theta}
cos⁡θ=1sec⁡θ\cos\theta = \frac{1}{\sec\theta}
sec⁡θ=1cos⁡θ\sec\theta = \frac{1}{\cos\theta}
tan⁡θ=1cot⁡θ\tan\theta = \frac{1}{\cot\theta}
cot⁡θ=1tan⁡θ\cot\theta = \frac{1}{\tan\theta}

2. Pythagorean Identities

Derived from the Pythagorean theorem:

sin⁡2θ+cos⁡2θ=1\sin^2\theta + \cos^2\theta = 1
1+tan⁡2θ=sec⁡2θ1 + \tan^2\theta = \sec^2\theta
1+cot⁡2θ=csc⁡2θ1 + \cot^2\theta = \csc^2\theta

3. Quotient Identities

Relate tangent and cotangent to sine and cosine:

tan⁡θ=sin⁡θcos⁡θ\tan\theta = \frac{\sin\theta}{\cos\theta}
cot⁡θ=cos⁡θsin⁡θ\cot\theta = \frac{\cos\theta}{\sin\theta}

4. Cofunction Identities

Show relationships between complementary angles:

sin⁡(90∘−θ)=cos⁡θ\sin(90^\circ - \theta) = \cos\theta
cos⁡(90∘−θ)=sin⁡θ\cos(90^\circ - \theta) = \sin\theta
tan⁡(90∘−θ)=cot⁡θ\tan(90^\circ - \theta) = \cot\theta
cot⁡(90∘−θ)=tan⁡θ\cot(90^\circ - \theta) = \tan\theta
sec⁡(90∘−θ)=csc⁡θ\sec(90^\circ - \theta) = \csc\theta
csc⁡(90∘−θ)=sec⁡θ\csc(90^\circ - \theta) = \sec\theta

5. Even-Odd Identities

Indicate symmetry of trig functions:

sin⁡(−θ)=−sin⁡θ\sin(-\theta) = -\sin\theta
cos⁡(−θ)=cos⁡θ\cos(-\theta) = \cos\theta
tan⁡(−θ)=−tan⁡θ\tan(-\theta) = -\tan\theta

6. Double-Angle Identities

sin⁡(2θ)=2sin⁡θcos⁡θ\sin(2\theta) = 2\sin\theta\cos\theta
cos⁡(2θ)=cos⁡2θ−sin⁡2θ\cos(2\theta) = \cos^2\theta - \sin^2\theta
tan⁡(2θ)=2tan⁡θ1−tan⁡2θ\tan(2\theta) = \frac{2\tan\theta}{1 - \tan^2\theta}

7. Half-Angle Identities

sin⁡θ2=±1−cos⁡θ2\sin\frac{\theta}{2} = \pm\sqrt{\frac{1 - \cos\theta}{2}}
cos⁡θ2=±1+cos⁡θ2\cos\frac{\theta}{2} = \pm\sqrt{\frac{1 + \cos\theta}{2}}
tan⁡θ2=sin⁡θ1+cos⁡θ\tan\frac{\theta}{2} = \frac{\sin\theta}{1 + \cos\theta} or 1−cos⁡θsin⁡θ\frac{1 - \cos\theta}{\sin\theta}

8. Sum & Difference Identities

sin⁡(α±β)=sin⁡αcos⁡β±cos⁡αsin⁡β\sin(\alpha \pm \beta) = \sin\alpha\cos\beta \pm \cos\alpha\sin\beta
cos⁡(α±β)=cos⁡αcos⁡β∓sin⁡αsin⁡β\cos(\alpha \pm \beta) = \cos\alpha\cos\beta \mp \sin\alpha\sin\beta
tan⁡(α±β)=tan⁡α±tan⁡β1∓tan⁡αtan⁡β\tan(\alpha \pm \beta) = \frac{\tan\alpha \pm \tan\beta}{1 \mp \tan\alpha\tan\beta}

Tips for Remembering All Trig Identities

Group them by type — Pythagorean, reciprocal, etc.
Memorize the core — Once you know sine, cosine, and tangent rules, you can derive the rest.
Practice regularly — Repetition through problem-solving cements recall.
Visualize on the unit circle — Many identities become obvious when seen geometrically.

FAQ – All Trigonometric Identities

1. Do I need to memorize every trig identity?
Yes — especially for timed AP exams and when calculators aren’t allowed.

2. Which identities are used most on AP exams?
Pythagorean, double-angle, and sum/difference identities appear most often.

3. How can I prove an identity?
Start with one side of the equation, simplify using known identities, and aim to match the other side.

4. Will these appear on the AP Calculus exam?
Yes — especially in integration and differentiation of trig functions.

5. Are trig identities on the formula sheet?
No — you’re expected to know them.

6. Can I derive them instead of memorizing?
Yes, but memorizing speeds up test performance.

7. Are these the same for radians and degrees?
Yes — trig identities work in both.

8. What’s the fastest way to learn them?
Flashcards, spaced repetition, and solving a variety of identity-proving problems.

Turn Your Math Skills Into a College Advantage

Strong trig identity mastery can help you score higher on AP Calc and SAT Math.

🎯 Take the University Match Quiz to find schools that value high math achievement.
🌍 Browse the Universities Directory for admissions data, rankings, and program details.

All Trigonometric Identities — Complete List & How to Use Them | RevisionDojo