Essential Cosine Identities for Trigonometry

What Cosine Identities Actually Are

Cosine identities are equations involving cosine that are always true. They let you rewrite, simplify, and solve trig problems without starting from scratch every time.

You need maybe 5 or 6 of these. The rest are variations. Learn the core ones and you can derive the rest.

The Pythagorean Identity

This is the foundation. Everything else builds from this:

cos²θ + sin²θ = 1

It comes straight from the unit circle. If you forget it during a test, draw a right triangle and you'll see why it works.

You can rearrange it to solve for either term:

• cos²θ = 1 - sin²θ
• sin²θ = 1 - cos²θ

That's useful when you need to convert between sine and cosine in an expression.

Double Angle Formulas for Cosine

When you need cos(2θ), you have three equivalent forms. Pick whichever fits your problem:

cos(2θ) = cos²θ - sin²θ

cos(2θ) = 2cos²θ - 1

cos(2θ) = 1 - 2sin²θ

The second and third forms are useful when you only have cosine or only have sine in your problem. Replace the other term using the Pythagorean identity.

When to Use Each Form

• cos²θ - sin²θ works when you have both terms
• 2cos²θ - 1 works when the problem gives you cos²θ
• 1 - 2sin²θ works when the problem gives you sin²θ

Sum and Difference Formulas

These let you break apart cosines of added or subtracted angles:

cos(α + β) = cos α cos β - sin α sin β

cos(α - β) = cos α cos β + sin α sin β

The sign on the second term flips. That's the only difference between the two formulas.

These are also reversible. If you have cos α cos β - sin α sin β, you can combine it back into cos(α + β). This comes up constantly in integration and solving trig equations.

The Law of Cosines

Not technically an "identity" but it uses cosine and appears on every exam:

c² = a² + b² - 2ab·cos(C)

Use this when you have two sides and the included angle, or all three sides and need to find an angle. It's just the Pythagorean theorem with a cosine correction term.

When C = 90°, cos(C) = 0 and the formula collapses to the regular Pythagorean theorem. That tells you it's working correctly.

Half Angle Formulas

When you need cos(θ/2), use one of these:

cos(θ/2) = ±√((1 + cos θ)/2)

The ± sign depends on which quadrant θ/2 lands in. Figure out the sign before you write your answer.

This formula comes from the double angle formula run backwards. Derive it during your test if you forget it: start with cos(2α) = 2cos²α - 1, set α = θ/2, and solve for cos(θ/2).

Cosine in Terms of Euler's Formula

If you're comfortable with complex numbers, cosine has a clean representation:

cos θ = (e^(iθ) + e^(-iθ))/2

This makes the sum formulas trivial to derive and explains why cosine appears in signal processing and wave equations. Not required reading for basic trig, but useful if you go further.

Quick Reference Table

How to Actually Use These

Here's a typical problem: simplify cos²θ - sin²θ

Step 1: Recognize this as a double angle formula. It equals cos(2θ). Done.

Another one: find cos(75°)

Step 1: Write 75° as 45° + 30°

Step 2: Apply the sum formula

cos(75°) = cos(45°)cos(30°) - sin(45°)sin(30°)

cos(75°) = (√2/2)(√3/2) - (√2/2)(1/2)

cos(75°) = (√6 - √2)/4

That's your answer. No guessing, just applying the formula.

What to Actually Memorize

You need these cold:

• cos²θ + sin²θ = 1
• cos(α ± β) formulas
• cos(2θ) in its three forms
• Law of cosines

Everything else you can derive from these. If a problem gives you something that looks unfamiliar, try rewriting it in terms of these core identities.