Trigonometric identities are equations that are true for all allowed angle values. Students need them to simplify expressions, solve equations, verify relationships, and connect graphs with the unit circle. This cheat sheet organizes the main identity families used in grades 10-12 so students can choose the right tool quickly.

The most important identities come from the unit circle and the relationship between sine, cosine, and tangent. Core ideas include rewriting functions with reciprocal and quotient identities, using sin⁡2θ+cos⁡2θ=1\sin^2 \theta + \cos^2 \theta = 1, and applying sum and difference formulas. Double-angle and half-angle identities help transform expressions involving 2θ2\theta or θ2\frac{\theta}{2} into easier forms.

Key Facts

  • The reciprocal identities are csc⁡θ=1sin⁡θ\csc \theta = \frac{1}{\sin \theta}, sec⁡θ=1cos⁡θ\sec \theta = \frac{1}{\cos \theta}, and cot⁡θ=1tan⁡θ\cot \theta = \frac{1}{\tan \theta}.
  • The quotient identities are tan⁡θ=sin⁡θcos⁡θ\tan \theta = \frac{\sin \theta}{\cos \theta} and cot⁡θ=cos⁡θsin⁡θ\cot \theta = \frac{\cos \theta}{\sin \theta}.
  • The main Pythagorean identity is sin⁡2θ+cos⁡2θ=1\sin^2 \theta + \cos^2 \theta = 1.
  • Dividing sin⁡2θ+cos⁡2θ=1\sin^2 \theta + \cos^2 \theta = 1 by cos⁡2θ\cos^2 \theta gives 1+tan⁡2θ=sec⁡2θ1 + \tan^2 \theta = \sec^2 \theta.
  • Dividing sin⁡2θ+cos⁡2θ=1\sin^2 \theta + \cos^2 \theta = 1 by sin⁡2θ\sin^2 \theta gives 1+cot⁡2θ=csc⁡2θ1 + \cot^2 \theta = \csc^2 \theta.
  • The sum and difference identities are sin⁡(a±b)=sin⁡acos⁡b±cos⁡asin⁡b\sin(a \pm b) = \sin a \cos b \pm \cos a \sin b and cos⁡(a±b)=cos⁡acos⁡b∓sin⁡asin⁡b\cos(a \pm b) = \cos a \cos b \mp \sin a \sin b.
  • The double-angle identities include sin⁡(2θ)=2sin⁡θcos⁡θ\sin(2\theta) = 2\sin \theta \cos \theta and cos⁡(2θ)=cos⁡2θ−sin⁡2θ\cos(2\theta) = \cos^2 \theta - \sin^2 \theta.
  • The half-angle identities are sin⁡2(θ2)=1−cos⁡θ2\sin^2 \left(\frac{\theta}{2}\right) = \frac{1 - \cos \theta}{2} and cos⁡2(θ2)=1+cos⁡θ2\cos^2 \left(\frac{\theta}{2}\right) = \frac{1 + \cos \theta}{2}.

Vocabulary

Trigonometric identity
A trigonometric identity is an equation involving trigonometric functions that is true for every angle where both sides are defined.
Reciprocal identity
A reciprocal identity rewrites a trigonometric function as the reciprocal of another function, such as sec⁡θ=1cos⁡θ\sec \theta = \frac{1}{\cos \theta}.
Quotient identity
A quotient identity expresses tangent or cotangent as a ratio, such as tan⁡θ=sin⁡θcos⁡θ\tan \theta = \frac{\sin \theta}{\cos \theta}.
Pythagorean identity
A Pythagorean identity connects squared trigonometric functions, with the main one being sin⁡2θ+cos⁡2θ=1\sin^2 \theta + \cos^2 \theta = 1.
Cofunction identity
A cofunction identity relates complementary angles, such as sin⁡θ=cos⁡(90∘−θ)\sin \theta = \cos(90^\circ - \theta).
Double-angle identity
A double-angle identity rewrites a function of 2θ2\theta using functions of θ\theta, such as sin⁡(2θ)=2sin⁡θcos⁡θ\sin(2\theta) = 2\sin \theta \cos \theta.

Common Mistakes to Avoid

  • Writing sin⁡2θ\sin^2 \theta as sin⁡(2θ)\sin(2\theta) is wrong because sin⁡2θ\sin^2 \theta means (sin⁡θ)2(\sin \theta)^2, not the sine of a doubled angle.
  • Changing the sign incorrectly in cos⁡(a±b)\cos(a \pm b) is wrong because cos⁡(a+b)=cos⁡acos⁡b−sin⁡asin⁡b\cos(a + b) = \cos a \cos b - \sin a \sin b and cos⁡(a−b)=cos⁡acos⁡b+sin⁡asin⁡b\cos(a - b) = \cos a \cos b + \sin a \sin b.
  • Canceling across sums such as sin⁡θ+1sin⁡θ\frac{\sin \theta + 1}{\sin \theta} is wrong because terms in a sum cannot be canceled separately unless they are common factors.
  • Using tan⁡θ=cos⁡θsin⁡θ\tan \theta = \frac{\cos \theta}{\sin \theta} is wrong because tan⁡θ=sin⁡θcos⁡θ\tan \theta = \frac{\sin \theta}{\cos \theta} and cot⁡θ=cos⁡θsin⁡θ\cot \theta = \frac{\cos \theta}{\sin \theta}.
  • Forgetting domain restrictions is wrong because identities like tan⁡θ=sin⁡θcos⁡θ\tan \theta = \frac{\sin \theta}{\cos \theta} are only valid where cos⁡θ≠0\cos \theta \ne 0.

Practice Questions

  1. 1 Simplify 1−cos⁡2θsin⁡θ\frac{1 - \cos^2 \theta}{\sin \theta} using a Pythagorean identity.
  2. 2 Find the exact value of sin⁡(75∘)\sin(75^\circ) using sin⁡(a+b)\sin(a + b).
  3. 3 Rewrite cos⁡(2θ)\cos(2\theta) in two different equivalent forms using Pythagorean identities.
  4. 4 Explain why proving an identity usually works better by transforming one side into the other instead of substituting one angle value.

Understanding Trigonometric Identities

An identity works like a translation rule. It changes the form of an expression without changing its value. This matters because one form may reveal a useful feature that another form hides.

For example, an expression written only with sine and cosine is often easier to combine, factor, or compare. A form involving tangent may be better when a problem contains ratios. Each rewrite has restrictions.

A fraction cannot have zero in its denominator, so a transformed expression may be undefined at certain angles. Keep those excluded angles in mind, especially when solving equations.

The unit circle explains why the main relationships fit together. A point on this circle has horizontal coordinate cosine and vertical coordinate sine. Its distance from the origin stays one unit.

The distance rule for horizontal and vertical movement creates the central squared relationship. The other Pythagorean forms come from dividing that relationship by a square of sine or cosine. Addition formulas have a different source.

They describe what happens when rotations are combined. Setting both angles equal in an addition formula produces double angle results.

This is why several forms of the cosine double angle relationship exist. They are equal, yet one version may be much more useful than another in a particular problem.

When verifying an identity, work on one side only. Choose the side that looks more complicated and rewrite it in small steps until it matches the other side. Do not begin by changing both sides, because two separate chains of work can hide an error.

Look for common patterns before choosing a formula. Fractions often suggest finding a common denominator. Powers of sine and cosine often suggest the Pythagorean relationship.

Products such as two sine times cosine can suggest a double angle rewrite. Factoring is important here.

Many expressions look unrelated until a common factor is pulled out. Every line should follow from one clear rule, not from a guess that the final result ought to work.

Half angle work needs extra care. A squared sine or cosine value gives a magnitude, not always the final signed value. The angle location determines whether sine or cosine is positive or negative.

This is a common source of lost marks. Students meet these ideas in graphing waves, finding exact values for unusual angles, resolving forces into horizontal and vertical parts, and studying periodic motion. A swinging pendulum, a rotating wheel, and an alternating electric signal can all be modeled with trigonometric functions.

Build fluency by learning the meaning behind each family, then practice recognizing patterns. Memorizing a long list without using it makes formulas easy to confuse.