What Are Trigonometric Identities?
A trigonometric identity is an equation involving trigonometric functions that is true for every value of the variable where both sides are defined. Unlike a trigonometric equation (which is true only for specific angle values), an identity holds universally.
For example, sin²θ + cos²θ = 1 is an identity because no matter what value you substitute for θ, the equation holds. In contrast, sin θ = 1/2 is an equation because it is only true for specific values of θ (like 30° and 150°).
Identities are the primary tools for simplifying expressions, verifying equivalences, and solving equations throughout trigonometry, precalculus, and calculus. Memorizing the core identities and understanding how to apply them is essential for success in any mathematics course that involves trigonometric functions.
The Pythagorean Identities
The three Pythagorean identities are arguably the most important identities in all of trigonometry. They are derived from the Pythagorean theorem applied to the unit circle.
Proof of sin²θ + cos²θ = 1
On the unit circle (a circle of radius 1 centered at the origin), any point is described by coordinates (cos θ, sin θ). Since the point lies on the circle x² + y² = 1, substituting gives:
This identity is the foundation from which the other two Pythagorean identities are derived.
Deriving the Other Two
Divide sin²θ + cos²θ = 1 by cos²θ:
Divide sin²θ + cos²θ = 1 by sin²θ:
Key Takeaway
All three Pythagorean identities come from one source: sin²θ + cos²θ = 1. If you remember this single identity, you can derive the other two by dividing by cos²θ or sin²θ.
Reciprocal Identities
The six trigonometric functions come in three reciprocal pairs. The reciprocal identities express this relationship:
sec θ = 1 / cos θ
cot θ = 1 / tan θ
These identities are definitional. Cosecant is defined as the reciprocal of sine, secant as the reciprocal of cosine, and cotangent as the reciprocal of tangent. They are invaluable for rewriting expressions involving csc, sec, and cot in terms of sin, cos, and tan, which often simplifies calculations significantly.
Quotient Identities
The quotient identities express tangent and cotangent as ratios of sine and cosine:
cot θ = cos θ / sin θ
These follow directly from the definitions of the trig functions on the unit circle, where sin θ = y, cos θ = x, and tan θ = y/x. The quotient identities are used constantly when simplifying expressions or converting everything to sines and cosines.
Co-function Identities
The co-function identities reveal the deep relationship between complementary trigonometric function pairs. Two angles are complementary if they sum to 90° (or π/2 radians).
tan θ = cot(90° − θ) cot θ = tan(90° − θ)
sec θ = csc(90° − θ) csc θ = sec(90° − θ)
The word "cosine" literally means "complement's sine," and this is exactly what the co-function identities express. In a right triangle with acute angles θ and (90° − θ), the sine of one angle equals the cosine of the other because they reference the same side ratios from different perspectives.
Even-Odd Identities
The even-odd identities describe how trig functions behave when the input angle is negated:
cos(−θ) = cos θ (even)
tan(−θ) = −tan θ (odd)
Cosine is the only even trig function. Sine, tangent, cosecant, cotangent, and secant's even-odd status follows from these three:
- csc(−θ) = −csc θ (odd, since csc = 1/sin)
- sec(−θ) = sec θ (even, since sec = 1/cos)
- cot(−θ) = −cot θ (odd, since cot = cos/sin)
These identities are especially useful in calculus when evaluating integrals of trig functions over symmetric intervals, and in simplifying expressions involving negative angles.
How to Verify Trigonometric Identities
Verifying an identity means showing that the left side equals the right side for all valid values of the variable. Here is a systematic approach:
Strategy 1: Work with One Side Only
Pick the more complicated side and transform it step by step until it matches the other side. Never perform operations on both sides simultaneously (that technique is for solving equations, not verifying identities).
Strategy 2: Convert Everything to Sine and Cosine
Replace tan, cot, sec, and csc with their definitions in terms of sin and cos. This often reveals cancellations and simplifications that were hidden by the shorthand notation.
Strategy 3: Factor and Simplify
Look for common factors, difference of squares, or other algebraic patterns. For example, sin²θ − cos²θ factors as (sin θ − cos θ)(sin θ + cos θ).
Strategy 4: Use Pythagorean Substitutions
Replace sin²θ with 1 − cos²θ (or vice versa) when you see a sum or difference involving squared trig functions. This is one of the most frequently used techniques.
Strategy 5: Multiply by a Conjugate
If you see (1 − sin θ) or (1 + cos θ) in a denominator, multiplying numerator and denominator by the conjugate often unlocks a Pythagorean identity simplification.
Key Takeaway
When verifying identities: work one side only, convert to sin/cos when stuck, look for Pythagorean substitutions, and try conjugate multiplication for stubborn fractions. Practice is the key to developing intuition for which strategy to apply.
Worked Examples
Example 1: Verify an Identity
Verify: tan²θ · cos²θ + cos²θ = 1
Step 1: Start with the left side. Factor out cos²θ.
cos²θ(tan²θ + 1)
Step 2: Apply the Pythagorean identity tan²θ + 1 = sec²θ.
cos²θ · sec²θ
Step 3: Replace sec²θ with 1/cos²θ.
cos²θ · (1/cos²θ) = 1
Step 4: The left side equals 1, which is the right side. ■
Example 2: Simplify a Trig Expression
Simplify: (sin θ · csc θ) / (tan θ)
Step 1: Convert to sin and cos.
(sin θ · 1/sin θ) / (sin θ/cos θ)
Step 2: Simplify the numerator. sin θ · 1/sin θ = 1.
1 / (sin θ/cos θ)
Step 3: Divide by a fraction = multiply by reciprocal.
1 · cos θ/sin θ = cos θ/sin θ = cot θ
Example 3: Prove sin²θ + cos²θ = 1 from the Unit Circle
Goal: Prove the fundamental Pythagorean identity using the unit circle definition.
Step 1: Define the unit circle as x² + y² = 1 (a circle of radius 1 centered at the origin).
Step 2: For any angle θ, the point on the unit circle is (cos θ, sin θ) by definition of sine and cosine on the unit circle.
Step 3: Since this point lies on the circle, it must satisfy the equation: (cos θ)² + (sin θ)² = 1.
Step 4: Therefore sin²θ + cos²θ = 1 for all θ. ■
Example 4: Verify Using Conjugate Multiplication
Verify: 1 / (1 + sin θ) = (1 − sin θ) / cos²θ
Step 1: Start with the left side. Multiply numerator and denominator by (1 − sin θ).
(1 · (1 − sin θ)) / ((1 + sin θ)(1 − sin θ))
Step 2: Expand the denominator using difference of squares.
(1 − sin θ) / (1 − sin²θ)
Step 3: Apply the Pythagorean identity: 1 − sin²θ = cos²θ.
(1 − sin θ) / cos²θ
Step 4: This matches the right side. ■
Practice Problems
Practice 1
Simplify: sec θ · sin θ / tan θ
Solution:
Convert to sin and cos:
(1/cos θ) · sin θ / (sin θ/cos θ)
= (sin θ/cos θ) · (cos θ/sin θ)
= 1
Practice 2
Verify: (1 − cos²θ)(1 + cot²θ) = 1
Solution:
Replace 1 − cos²θ with sin²θ (Pythagorean identity):
sin²θ · (1 + cot²θ)
Replace 1 + cot²θ with csc²θ (Pythagorean identity):
sin²θ · csc²θ = sin²θ · (1/sin²θ) = 1 ■
Practice 3
Simplify: (sin θ + cos θ)² + (sin θ − cos θ)²
Solution:
Expand the first square: sin²θ + 2 sin θ cos θ + cos²θ
Expand the second square: sin²θ − 2 sin θ cos θ + cos²θ
Add them: 2 sin²θ + 2 cos²θ = 2(sin²θ + cos²θ) = 2 · 1 = 2
Practice 4
Verify: tan θ + cot θ = sec θ · csc θ
Solution:
Convert the left side to sin/cos:
sin θ/cos θ + cos θ/sin θ
Common denominator (sin θ cos θ):
(sin²θ + cos²θ) / (sin θ cos θ)
Apply Pythagorean identity:
1 / (sin θ cos θ) = (1/sin θ)(1/cos θ) = csc θ · sec θ = sec θ · csc θ ■