Core Trigonometric Identities: Pythagorean and Simple Ratio Relationships

What an Identity Is (and How to Use One)

An identity is an equation that is true for every angle where both sides are defined. You do not “solve” an identity for a particular angle; instead, you use it to rewrite expressions into equivalent forms. In practice, identities help you simplify complicated expressions, verify that two expressions match, and compute exact values efficiently.

The Pythagorean Identity from the Unit Circle

On the unit circle, a point at angle θ has coordinates (cos θ, sin θ). Because every point on the unit circle is exactly 1 unit from the origin, it satisfies the circle equation x² + y² = 1.

Substitute x = cos θ and y = sin θ:

(cos θ)² + (sin θ)² = 1

Rewriting in the common order gives the Pythagorean identity:

sin²θ + cos²θ = 1

Useful rearrangements

Often you need to isolate one squared trig function:

Continue in our app.

You can listen to the audiobook with the screen off, receive a free certificate for this course, and also have access to 5,000 other free online courses.

Or continue reading below...

Download the app

• sin²θ = 1 − cos²θ

• cos²θ = 1 − sin²θ

These are still identities; they are just algebraic rearrangements of the same always-true relationship.

Quotient and Reciprocal Identities (Simple Ratio Relationships)

Quotient identities

These come directly from the definition of tangent (and cotangent) as ratios of sine and cosine:

tan θ = sin θ / cos θ     (requires cos θ ≠ 0)

cot θ = cos θ / sin θ     (requires sin θ ≠ 0)

Reciprocal identities

These define secant and cosecant as reciprocals of cosine and sine, and relate cotangent to tangent:

sec θ = 1 / cos θ     (requires cos θ ≠ 0)

csc θ = 1 / sin θ     (requires sin θ ≠ 0)

cot θ = 1 / tan θ     (requires tan θ ≠ 0)

When simplifying, it is usually safest to convert everything into sin and cos first, because sin²θ + cos²θ = 1 only involves those two functions.

Practice Set A: Verify an Identity by Rewriting One Side

Verification strategy: pick the more complicated side, rewrite it using quotient/reciprocal identities and algebra, and aim to match the other side. Avoid manipulating both sides at once; it is easy to accidentally “prove” something by doing an invalid step on both sides.

Example 1: Verify sec²θ − tan²θ = 1

Goal: show the left side simplifies to 1.

Step 1: Rewrite in sin and cos.

sec²θ − tan²θ = (1/cos²θ) − (sin²θ/cos²θ)

Step 2: Combine over a common denominator.

(1 − sin²θ) / cos²θ

Step 3: Use the Pythagorean identity in rearranged form.

1 − sin²θ = cos²θ

Step 4: Substitute and simplify.

(cos²θ) / cos²θ = 1

This verifies the identity (for angles where cos θ ≠ 0, since sec and tan must be defined).

Example 2: Verify (1 − cos²θ)/sin²θ = 1

Step 1: Start with the left side.

(1 − cos²θ)/sin²θ

Step 2: Replace 1 − cos²θ using sin²θ = 1 − cos²θ.

sin²θ / sin²θ

Step 3: Simplify.

1

This is valid wherever sin θ ≠ 0 (because the original expression divides by sin²θ).

Practice Set B: Simplify by Converting Everything to sin and cos

Common approach: (1) rewrite sec, csc, tan, cot using sin and cos, (2) combine fractions, (3) use sin²θ + cos²θ = 1 if it appears.

Example 3: Simplify (1 − cos²θ)/cos θ

Step 1: Use 1 − cos²θ = sin²θ.

(1 − cos²θ)/cos θ = sin²θ / cos θ

Step 2: Decide whether to rewrite further. You can leave it as sin²θ/cos θ, or express part of it as tan θ:

sin²θ / cos θ = sin θ · (sin θ / cos θ) = sin θ · tan θ

Both are equivalent where cos θ ≠ 0.

Example 4: Simplify (sec θ)(cos θ) + (tan θ)(cos θ)

Step 1: Convert to sin and cos.

(sec θ)(cos θ) + (tan θ)(cos θ) = (1/cos θ)(cos θ) + (sin θ/cos θ)(cos θ)

Step 2: Cancel factors.

1 + sin θ

This simplification is valid where cos θ ≠ 0 (because sec and tan require it).

Example 5: Simplify (csc θ − cot θ)(csc θ + cot θ)

Step 1: Recognize a difference of squares pattern.

(csc θ − cot θ)(csc θ + cot θ) = csc²θ − cot²θ

Step 2: Convert to sin and cos.

csc²θ − cot²θ = (1/sin²θ) − (cos²θ/sin²θ)

Step 3: Combine and use the Pythagorean identity.

(1 − cos²θ)/sin²θ = sin²θ/sin²θ = 1

Valid where sin θ ≠ 0.

Practice Set C: Evaluate Expressions at Special Angles (Exact Values)

When evaluating, use exact values for sin and cos, then build tan, sec, csc, cot from quotient/reciprocal identities. Keep answers exact (fractions and radicals), not decimals.

Example 6: Evaluate sec(π/3) + tan(π/3)

Step 1: Use sin and cos at π/3.

• cos(π/3) = 1/2

• sin(π/3) = √3/2

Step 2: Build sec and tan.

sec(π/3) = 1 / cos(π/3) = 1 / (1/2) = 2

tan(π/3) = sin(π/3) / cos(π/3) = (√3/2) / (1/2) = √3

Step 3: Add.

sec(π/3) + tan(π/3) = 2 + √3

Example 7: Evaluate (1 − sin²(π/6)) / cos²(π/6)

Step 1: Use sin(π/6) and cos(π/6).

• sin(π/6) = 1/2, so sin²(π/6) = 1/4

• cos(π/6) = √3/2, so cos²(π/6) = 3/4

Step 2: Substitute and simplify.

(1 − 1/4) / (3/4) = (3/4) / (3/4) = 1

Mini-Quiz (Show Clear Steps)

Instructions: For each problem, write the required steps. For verification, rewrite one side only. For simplification, convert to sin and cos. For evaluation, start from sin and cos at the given angle.

1) Verify an identity

Verify: (1 + tan²θ)cos²θ = 1

• Step requirement: Rewrite tan²θ as sin²θ/cos²θ, then simplify using sin²θ + cos²θ = 1.

2) Verify an identity

Verify: (sec θ − 1)(sec θ + 1) = tan²θ

• Step requirement: Use difference of squares on the left, then rewrite sec²θ and tan²θ in sin/cos and use sin²θ + cos²θ = 1.

3) Simplify an expression

Simplify: (sin θ)(sec θ) − (cos θ)(tan θ)

• Step requirement: Convert sec and tan to 1/cos and sin/cos, combine terms over a common denominator, and simplify.

4) Simplify an expression

Simplify: (1 − cos²θ)/(sin θ)

• Step requirement: Replace 1 − cos²θ with sin²θ, then reduce the fraction.

5) Evaluate exactly

Evaluate: csc(π/4) − cot(π/4)

• Step requirement: Start with sin(π/4) and cos(π/4), then compute csc = 1/sin and cot = cos/sin.

6) Evaluate exactly

Evaluate: (tan(π/6))(sec(π/6))

• Step requirement: Use sin(π/6) and cos(π/6) to compute tan and sec, then multiply and simplify radicals if needed.