Mastering Trigonometric Identities- A Complete Guide
What Trigonometric Identities Actually Are
Trigonometric identities are equations that hold true for every possible value of the variable. That's the whole deal. They describe relationships between sine, cosine, tangent, and their reciprocals.
You need these identities for three main reasons:
- Simplifying expressions
- Solving equations
- Evaluating integrals and limits
If you're taking calculus, physics, or engineering, you'll use these constantly. No way around it.
The Core Identities You Must Know
Pythagorean Identities
These come straight from the Pythagorean theorem applied to the unit circle. Memorize them first.
- sin²θ + cos²θ = 1 — the foundation of everything
- 1 + tan²θ = sec²θ — derived from the first one
- 1 + cot²θ = csc²θ — same deal
These three will appear in almost every trig problem you encounter. If you forget one, you can derive it from sin²θ + cos²θ = 1.
Reciprocal Identities
Simple relationships between the six trig functions:
- csc θ = 1/sin θ
- sec θ = 1/cos θ
- cot θ = 1/tan θ
- tan θ = sin θ/cos θ
- cot θ = cos θ/sin θ
Most people know these. The problem is remembering which goes where when you're in the middle of a problem.
Sum and Difference Formulas
These let you break down angles that aren't "nice" values.
sin(A ± B) = sin A cos B ± cos A sin B
cos(A ± B) = cos A cos B ∓ sin A sin B
tan(A ± B) = (tan A ± tan B) / (1 ∓ tan A tan B)
The ± and ∓ symbols mean the top signs match and the bottom signs match. If you use + in the numerator, use - in the denominator.
Quick Example
Find sin(75°).
75° = 45° + 30°
sin(75°) = sin(45°)cos(30°) + cos(45°)sin(30°)
= (√2/2)(√3/2) + (√2/2)(1/2)
= (√6 + √2)/4
That's the answer. No calculator required.
Double Angle Formulas
These are special cases of the sum formulas where both angles are equal.
sin(2θ) = 2 sin θ cos θ
cos(2θ) = cos²θ - sin²θ
This one has alternate forms that are often more useful:
- cos(2θ) = 2cos²θ - 1
- cos(2θ) = 1 - 2sin²θ
tan(2θ) = 2 tan θ / (1 - tan²θ)
Use the alternate cos(2θ) forms depending on what you're trying to eliminate. If you have sin²θ, use 1 - 2sin²θ. If you have cos²θ, use 2cos²θ - 1.
Half Angle Formulas
Flip the double angle formulas around. You get:
sin(θ/2) = ±√((1 - cos θ)/2)
cos(θ/2) = ±√((1 + cos θ)/2)
tan(θ/2) = ±√((1 - cos θ)/(1 + cos θ))
The ± sign depends on which quadrant θ/2 lands in. Figure out the sign before you write your final answer.
There's also a useful form for tan(θ/2) that avoids radicals:
tan(θ/2) = (1 - cos θ)/sin θ = sin θ/(1 + cos θ)
Product-to-Sum and Sum-to-Product
These convert between products and sums. Useful for integration and solving equations.
Product-to-Sum
- sin A cos 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
- 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)
Cofunction Identities
These describe how trig functions relate to their "co-" counterparts (sine and cosine flip, tangent and cotangent flip, secant and cosecant flip).
- 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.
The general rule: cofunction(θ) = function(π/2 - θ)
Even-Odd Identities
Determines symmetry behavior:
- sin(-θ) = -sin θ — odd function
- cos(-θ) = cos θ — even function
- tan(-θ) = -tan θ — odd function
Sine and tangent flip sign when you negate the angle. Cosine doesn't change.
Identities Reference Table
| Category | Identity |
|---|---|
| Pythagorean | sin²θ + cos²θ = 1 |
| 1 + tan²θ = sec²θ | |
| 1 + cot²θ = csc²θ | |
| Double Angle | sin(2θ) = 2 sin θ cos θ |
| cos(2θ) = cos²θ - sin²θ | |
| Sum/Difference | sin(A+B) = sin A cos B + cos A sin B |
| cos(A+B) = cos A cos B - sin A sin B |
How to Actually Use These: Getting Started
Knowing the identities isn't enough. You need to know when to use which one.
Step 1: Identify What You're Working With
Look at the expression or equation. Are you seeing:
- Powers of sin or cos? → Pythagorean identities
- Sum of angles? → Sum/difference formulas
- Twice an angle? → Double angle formulas
- Half an angle? → Half angle formulas
Step 2: Convert Everything to Sine and Cosine
If you're stuck, rewrite tan, sec, csc, and cot in terms of sin and cos. This almost always opens a path forward.
Step 3: Look for Patterns
Factor where possible. Group terms that look like the left side of an identity. The goal is usually to match one of the standard forms.
Step 4: Check Your Work
Pick a value (like θ = 30° or π/6) and verify both sides give the same result. If they don't, you made a mistake somewhere.
Common Mistakes That Cost Points
- Confusing addition formulas with multiplication. sin(A+B) ≠ sin A + sin B. That's just wrong.
- Dropping the ± on half-angle formulas. The sign matters. Always.
- Forgetting the quotient identity tan = sin/cos. This is often the key step to simplifying complex expressions.
- Using the wrong Pythagorean identity. sec²θ - tan²θ = 1, not sec²θ + tan²θ = 1.
When to Derive vs. Memorize
Some identities you should have memorized cold:
- sin²θ + cos²θ = 1
- sin(2θ) and cos(2θ)
- sin(A±B) and cos(A±B)
Others you can derive on the fly:
- Product-to-sum formulas (easy to derive from sum formulas)
- cot²θ = csc²θ - 1 (just rearrange the Pythagorean identity)
If you're in an exam and can't remember an obscure identity, derive it from the basics. It'll take 30 seconds and you'll get full credit.