Trigonometric Identities

Pythagorean, quotient, reciprocal, and algebraic trig identities.

A trigonometric identity is an equation that is true for every angle where both sides are defined. Identities let us rewrite expressions, simplify formulas, and prepare for calculus. They are not just things to memorize; they come from the unit circle and from definitions.

Learning objectives

After this lesson you should be able to:

  • use the Pythagorean identity;
  • rewrite tangent as sine over cosine;
  • use reciprocal identities;
  • verify simple identities step by step;
  • avoid illegal algebra such as dividing by zero;
  • use SymPy to check identity transformations.

What is an identity?

An equation such as

sin^2(x) + cos^2(x) = 1

is an identity because it is true for all angles x. An equation such as sin(x)=1/2 is not an identity; it is true only for certain angles.

Pythagorean identity

On the unit circle, the point is (cos(x), sin(x)). Since the radius is 1:

cos^2(x) + sin^2(x) = 1

Usually this is written:

sin^2(x) + cos^2(x) = 1

Rearranged forms are also useful:

sin^2(x) = 1 - cos^2(x)
cos^2(x) = 1 - sin^2(x)

Quotient identity

Tangent is sine divided by cosine:

tan(x) = sin(x) / cos(x)

This is defined only when cos(x) != 0.

Reciprocal identities

The reciprocal trig functions are:

csc(x) = 1 / sin(x)
sec(x) = 1 / cos(x)
cot(x) = 1 / tan(x) = cos(x) / sin(x)

They are undefined when their denominators are zero.

Verifying identities

To verify an identity, transform one side until it matches the other side. A safe strategy is:

  1. Start with the more complicated side.
  2. Rewrite everything using sine and cosine.
  3. Use sin^2(x)+cos^2(x)=1 when possible.
  4. Do ordinary algebra carefully.

Worked example 1: simplify

Simplify:

1 - sin^2(x)

Using the Pythagorean identity:

sin^2(x) + cos^2(x) = 1

Subtract sin^2(x) from both sides:

cos^2(x) = 1 - sin^2(x)

So:

1 - sin^2(x) = cos^2(x)

Worked example 2: verify an identity

Verify:

tan(x)cos(x) = sin(x)

Start with the left side:

tan(x)cos(x)
= (sin(x)/cos(x))cos(x)
= sin(x)

This matches the right side, for angles where cos(x) != 0.

Common pitfalls

  • Treating an identity like an equation with only a few solutions.
  • Dividing by sin(x) or cos(x) without considering when it is zero.
  • Changing both sides in unrelated ways.
  • Forgetting squared notation: sin^2(x) means (sin(x))^2.
  • Trying to prove an identity by checking only one angle.

Practice exercises

  1. Complete: sin^2(x)+cos^2(x)=?
  2. Rewrite tan(x) using sine and cosine.
  3. Rewrite sec(x) using cosine.
  4. Simplify 1 - cos^2(x).
  5. Verify cot(x)sin(x)=cos(x).
  1. 1.
  2. sin(x)/cos(x).
  3. 1/cos(x).
  4. sin^2(x).
  5. cot(x)sin(x) = (cos(x)/sin(x))sin(x)=cos(x) where sin(x) != 0.

Subtopic guided practice and checkpoints

Identity work is algebra with trig building blocks. Rewrite first; simplify second.

Lab 1: recognize an identity

Guess first. Is sin(x)=0 an identity?

Guided exercise.

No. It is true at some angles, such as 0 and pi, but not all angles. sin^2(x)+cos^2(x)=1 is true for all angles.

Checkpoint. Is tan(x)=sin(x)/cos(x) an identity? State any restriction.

Lab 2: use the Pythagorean identity

Guess first. What can replace 1 - sin^2(x)?

Guided exercise.

From sin^2(x)+cos^2(x)=1:

1 - sin^2(x) = cos^2(x)

Checkpoint. Replace 1 - cos^2(x) with a simpler expression.

Lab 3: rewrite with sine and cosine

Guess first. Why is rewriting tangent often helpful?

Guided exercise.

Rewrite:

tan(x)/sec(x)
= (sin(x)/cos(x)) / (1/cos(x))
= sin(x)

Checkpoint. Rewrite sec(x)cos(x) and simplify.

Lab 4: verify safely

Guess first. Should you start with the simpler side or the more complicated side?

Guided exercise.

Start with the more complicated side. To verify sec(x)-cos(x)=sin(x)tan(x), start left:

sec(x)-cos(x)
= 1/cos(x) - cos(x)
= (1 - cos^2(x))/cos(x)
= sin^2(x)/cos(x)
= sin(x)tan(x)

Checkpoint. Verify sin(x)cot(x)=cos(x).

  1. Yes, tan(x)=sin(x)/cos(x) where cos(x) != 0.
  2. sin^2(x).
  3. sec(x)cos(x)=(1/cos(x))cos(x)=1, where cos(x) != 0.
  4. sin(x)cot(x)=sin(x)(cos(x)/sin(x))=cos(x), where sin(x) != 0.

Identity guessing game

Using this lesson with edumath and SymPy

Use SymPy to simplify and verify identities.

import sympy as sp

x = sp.symbols("x")
sp.simplify(1 - sp.sin(x) ** 2)

\(\displaystyle \cos^{2}{\left(x \right)}\)

Check whether two sides simplify to zero.

left = sp.tan(x) * sp.cos(x)
right = sp.sin(x)
sp.simplify(left - right)

\(\displaystyle 0\)