Complete Trigonometric Identities Guide

What Trigonometric Identities Actually Are

Trigonometric identities are equations that hold true for every value where both sides are defined. That's the whole definition. They're not tricks or shortcuts—they're mathematical facts rooted in how triangles and circles relate to each other.

If you're studying calculus, physics, or engineering, you'll run into these constantly. The bad news: there's a lot of them. The good news: you only need to memorize the core ones. Everything else derives from those.

The Pythagorean Identities (The Foundation)

These come straight from the Pythagorean theorem applied to the unit circle. Memorize these first—they're the most useful.

• sin²θ + cos²θ = 1
• 1 + tan²θ = sec²θ
• 1 + cot²θ = csc²θ

The first one is the most important. If you forget everything else, remember that one.

Deriving the Other Two

You get 1 + tan²θ = sec²θ by dividing the first identity by cos²θ. Same logic for cot² and csc²—divide by sin²θ instead. This is useful when you're asked to "prove" an identity. Start with sin²θ + cos²θ = 1 and manipulate from there.

Reciprocal Identities

These connect the three main trig functions with their reciprocals.

• csc θ = 1/sin θ
• sec θ = 1/cos θ
• cot θ = 1/tan θ

Also useful: tan θ = sin θ/cos θ and cot θ = cos θ/sin θ

These look obvious written out, but they become essential when simplifying expressions. When you see 1/sin, write csc. When you see sin/cos, write tan. The notation swap makes algebra way cleaner.

Even and Odd Function Properties

This tells you what happens when you negate the angle.

• sin(−θ) = −sin θ (odd function)
• cos(−θ) = cos θ (even function)
• tan(−θ) = −tan θ (odd function)

Cotangent, cosecant, and secant follow the same pattern as their base functions. Cosine is the only "nice" one—it's unchanged by a sign flip.

Co-Function Identities

These relate trig functions of complementary angles (angles that add to π/2 or 90°).

• sin(π/2 − θ) = cos θ
• cos(π/2 − θ) = sin θ
• tan(π/2 − θ) = cot θ

In degrees: sin(90° − θ) = cos θ, and so on.

These identities are why sine and cosine are called "co-functions." The "co" prefix means they complement each other at right angles.

Sum and Difference Formulas

These let you break apart angles that don't have clean reference angles.

Sine of a Sum or Difference

• sin(A + B) = sin A cos B + cos A sin B
• sin(A − B) = sin A cos B − cos A sin B

Cosine of a Sum or Difference

• cos(A + B) = cos A cos B − sin A sin B
• cos(A − B) = cos A cos B + sin A sin B

Tangent of a Sum or Difference

• tan(A + B) = (tan A + tan B) / (1 − tan A tan B)
• tan(A − B) = (tan A − tan B) / (1 + tan A tan B)

The sign pattern matters. For sine: same sign (sin cos + cos sin). For cosine: different signs (cos cos − sin sin). Get this wrong and everything downstream breaks.

Double Angle Formulas

These are just the sum formulas where both angles are equal (A = B).

Sine Double Angle

• sin(2θ) = 2 sin θ cos θ

Cosine Double Angle (Three Forms)

• cos(2θ) = cos²θ − sin²θ
• cos(2θ) = 2cos²θ − 1
• cos(2θ) = 1 − 2sin²θ

Use the second form when you know cosine but not sine. Use the third when you know sine but not cosine. The first form is the default.

Tangent Double Angle

• tan(2θ) = 2 tan θ / (1 − tan²θ)

Half Angle Formulas

Flip the double angle formulas around. These come up in integration a lot.

• sin(θ/2) = ±√((1 − cos θ) / 2)
• cos(θ/2) = ±√((1 + cos θ) / 2)
• tan(θ/2) = ±√((1 − cos θ) / (1 + cos θ))

The ± is critical. The sign depends on which quadrant the half-angle lands in. If θ/2 is in the first or second quadrant, sine is positive. If it's in the third or fourth, sine is negative. Same logic for cosine and tangent.

Product-to-Sum Formulas

Convert products of sines and cosines into sums. Useful when you need to integrate products.

• sin A cos B = ½[sin(A + B) + sin(A − B)]
• cos A sin B = ½[sin(A + B) − sin(A − B)]
• cos A cos B = ½[cos(A + B) + cos(A − B)]
• sin A sin B = ½[cos(A − B) − cos(A + B)]

Sum-to-Product Formulas

The reverse. Convert sums into products.

• sin A + sin B = 2 sin((A + B)/2) cos((A − B)/2)
• sin A − sin B = 2 cos((A + B)/2) sin((A − B)/2)
• cos A + cos B = 2 cos((A + B)/2) cos((A − B)/2)
• cos A − cos B = −2 sin((A + B)/2) sin((A − B)/2)

Quick Reference Table

How to Use These Identities: A Practical Walkthrough

Simplifying an Expression

Example: Simplify (sin²θ)/(1 − cos²θ)

Step 1: Recognize that 1 − cos²θ = sin²θ (rearranged Pythagorean identity)

Step 2: (sin²θ)/(sin²θ) = 1

Done. That's it. The trick is recognizing which identity applies. That comes with practice, not memorization shortcuts.

Proving an Identity

The standard approach:

1. Convert everything to sin and cos (usually the safest starting point)
2. Look for patterns that match Pythagorean identities
3. Combine fractions if you see addition or subtraction
4. If stuck, try converting both sides to the same expression

Example: Prove tan θ + cot θ = sec θ csc θ

Left side: tan θ + cot θ = sin θ/cos θ + cos θ/sin θ = (sin²θ + cos²θ)/(sin θ cos θ) = 1/(sin θ cos θ)

Right side: sec θ csc θ = (1/cos θ)(1/sin θ) = 1/(sin θ cos θ)

Both sides match. Proved.

Solving Equations Using Identities

Example: Solve sin(2θ) = √3 cos θ for 0 ≤ θ < 2π

Step 1: Replace sin(2θ) with 2 sin θ cos θ

2 sin θ cos θ = √3 cos θ

Step 2: Move everything to one side

2 sin θ cos θ − √3 cos θ = 0

Step 3: Factor out cos θ

cos θ (2 sin θ − √3) = 0

Step 4: Set each factor to zero

• cos θ = 0 → θ = π/2, 3π/2
• 2 sin θ − √3 = 0 → sin θ = √3/2 → θ = π/3, 2π/3

Solutions: θ = π/2, 3π/2, π/3, 2π/3

Common Mistakes to Avoid

• Confusing addition formulas with double angle formulas. They're related but not interchangeable.
• Ignoring the ± in half-angle formulas. The square root doesn't give you a sign for free.
• Forgetting that tan, sec, csc, and cot are undefined at certain angles. cos θ = 0 means tan and sec are undefined. sin θ = 0 means cot and csc are undefined.
• Mixing up degrees and radians in the same problem. Pick one and stick with it.

What to Actually Memorize

You don't need to memorize all 20+ identities listed here. Focus on these 5 core ones:

1. sin²θ + cos²θ = 1
2. sin(A ± B) and cos(A ± B) formulas
3. sin(2θ) = 2 sin θ cos θ
4. cos(2θ) = cos²θ − sin²θ
5. tan θ = sin θ / cos θ

Everything else you can derive from these. If you're in a test situation and can't remember a formula, build it from the basics. It's slower but better than guessing wrong.