Trig Identities — MAT 172

A trig identity isn't something you memorize and forget — it's a relationship that's always true. The Pythagorean identity isn't a formula. It's just the unit circle. sin²θ + cos²θ = 1 because sin and cos are the sides of a right triangle inside a circle of radius 1.

The most important identities in trig. All three come directly from the unit circle and the Pythagorean theorem. You should be able to derive them, not just memorize them.

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Deriving identities 2 and 3 from identity 1

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You only need to memorize identity 1: sin²θ + cos²θ = 1. The other two are just algebra.

For identity 2 — divide every term by cos²θ:
sin²θ/cos²θ + cos²θ/cos²θ = 1/cos²θ
→ tan²θ + 1 = sec²θ ✓

For identity 3 — divide every term by sin²θ:
sin²θ/sin²θ + cos²θ/sin²θ = 1/sin²θ
→ 1 + cot²θ = csc²θ ✓

💡 On an exam you may be asked to "verify" or "prove" an identity — that means showing one side equals the other through algebraic manipulation. Never cross-multiply or move terms across the equals sign. Work on ONE side only.

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Pythagorean identity on the unit circle — live proof

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Pick any point on the unit circle. That point is (cos θ, sin θ). The distance from the origin to that point is always 1 (the radius). By the distance formula: √(cos²θ + sin²θ) = 1. Square both sides: cos²θ + sin²θ = 1. That's it. The identity isn't a formula — it's a geometric fact about every point on a circle of radius 1. It's true for every angle, every time, forever. Not because someone decided it — because it's the definition of what a unit circle is.

The reciprocal identities define the three less common trig functions — csc, sec, cot — as flips of the three main ones. If you know sin, cos, tan you automatically know these.

csc θ = 1 / sin θ

cosecant ▼

Cosecant is the reciprocal of sine. If sin θ = 1/2, then csc θ = 2. Note: csc is undefined when sin = 0 (at 0°, 180°, 360°). On the unit circle, csc θ is how far the laser tip would have to extend vertically to reach 1 — the reciprocal of how high it actually is.

sec θ = 1 / cos θ

secant ▼

Secant is the reciprocal of cosine. If cos θ = √3/2, then sec θ = 2/√3 = 2√3/3. Sec is undefined when cos = 0 (at 90° and 270°). Secant appears frequently in calculus — the derivative of tan θ is sec²θ, which connects directly to Pythagorean identity 2.

cot θ = 1 / tan θ

cotangent ▼

Cotangent is the reciprocal of tangent — and also equals cos θ / sin θ. Cot is undefined when tan = 0, which means when sin = 0 (at 0°, 180°). It's also equal to cos/sin, making it the quotient of the main two functions in the opposite order from tan.

Quotient identities express tan and cot as ratios of sin and cos. Only two to know, and they follow directly from SOHCAHTOA.

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The most common quotient identity mistake

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Students often write tan θ = cos θ / sin θ — that's cot, not tan. The easy memory trick: T comes before C in the alphabet, and tan = sin/cos (sin comes before cos alphabetically too). Or just remember: tan = the slope-like thing (rise over run = sin over cos).

✗ tan θ = cos/sin (WRONG — that's cot)
✓ tan θ = sin/cos
✓ cot θ = cos/sin

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Using quotient identities to simplify

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Whenever you see tan or cot in an expression, you can substitute sin/cos or cos/sin. This is useful for simplifying complex expressions — often converting everything to sin and cos makes the algebra cleaner.

Example: Simplify sin θ · cot θ
= sin θ · (cos θ / sin θ)
= cos θ ✓ (sin θ cancels)

Co-function identities relate each trig function to its complement. The "co" in cosine, cotangent, cosecant literally means complementary.

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The "co" means complement — etymology trick

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The prefix "co" in cosine, cotangent, and cosecant literally means complementary. Cosine = complement's sine. Cotangent = complement's tangent. Cosecant = complement's secant. So cos(30°) = sin(60°) because 60° is the complement of 30°. This is not a coincidence — it's built into the naming convention from the beginning. The "co" functions are literally the main functions of the complementary angle.

💡 Quick check: sin(30°) = 1/2 and cos(60°) = 1/2. Same value. Because 30° and 60° are complementary.

Watch sin²θ + cos²θ = 1 hold true at every angle. Drag the slider — both sides always equal 1. This is what "identity" means: true for every value of θ.

Angle θ 47°

sin²θ + cos²θ

1.000

sin²(47°) + cos²(47°)

= 1 always

1.000

✓ Identity holds

The bar never changes — sin² and cos² always add to exactly 1

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How to verify/prove an identity on an exam

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Rule 1: Work on ONE side only. Pick the more complicated side and simplify it to match the other. Never move terms across the equals sign.

Rule 2: Convert to sin and cos first. Replace tan, cot, sec, csc with their sin/cos equivalents.

Rule 3: Look for Pythagorean identity opportunities. Any time you see 1 − sin²θ, replace it with cos²θ. Any time you see sin²θ + cos²θ, replace it with 1.

Rule 4: Factor when possible. sin²θ − cos²θ = (sin θ + cos θ)(sin θ − cos θ).

💡 If you're totally stuck — try multiplying numerator and denominator by the conjugate, or expand a squared term.

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Worked example — verify an identity

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Verify: (1 − cos²θ) / sin θ = sin θ

Work on the left side only.

Step 1: Recognize 1 − cos²θ = sin²θ (Pythagorean identity rearranged)
→ sin²θ / sin θ

Step 2: Cancel one sin θ
→ sin θ ✓

Left side = right side. Identity verified. Done in 2 steps because we recognized the Pythagorean identity immediately.

Identity cheat sheet

Pythagorean

mainsin²θ + cos²θ = 1

÷cos²1 + tan²θ = sec²θ

÷sin²cot²θ + 1 = csc²θ

Quotient

tansin θ / cos θ

cotcos θ / sin θ

Reciprocal

csc1 / sin θ

sec1 / cos θ

cot1 / tan θ

Key vocabulary

Trig identity

An equation involving trig functions that is true for ALL values of the variable (where defined). Different from an equation that's only true for specific angles.

Pythagorean identity

sin²θ + cos²θ = 1 — and its two derived forms. Comes directly from the unit circle and Pythagorean theorem.

Reciprocal identity

csc = 1/sin, sec = 1/cos, cot = 1/tan. The three "co" functions are reciprocals of the three main functions.

Co-function identity

sin(θ) = cos(90°−θ) and related pairs. "Co" literally means complement — cosine is the complement's sine.

Verify an identity

Show algebraically that both sides of an equation are equal. Work on ONE side only — never move terms across the equals sign.

Exam reminders

Only memorize sin²θ + cos²θ = 1. Derive the other two by dividing by cos²θ or sin²θ.

When verifying identities: work on ONE side only. Convert everything to sin and cos first.

Any time you see 1 − sin²θ — that equals cos²θ. Instant substitution.

tan = sin/cos always. Not cos/sin — that's cot. Easy to flip on exams.

The "co" functions (cosine, cotangent, cosecant) are the complement's versions of the main functions. cos(30°) = sin(60°).

Reciprocals are undefined when the denominator = 0. tan and sec undefined at 90°/270°. cot and csc undefined at 0°/180°.