Fundamental Trigonometric Identities Guide
What Trigonometric Identities Actually Are
Trigonometric identities are equations that hold true for every possible value of the variable. They're not formulas you apply once and forget. They're relationships between trig functions that always work, no matter what angle you're plugging in.
Most students memorize these without understanding them. That's a mistake. If you know why these hold, you can derive them on the fly when your memory fails.
The Foundation: Pythagorean Identities
These come directly from the Pythagorean theorem applied to a right triangle. Every other identity branches from here.
- sin²θ + cos²θ = 1 — the most important one. Memorize it first.
- 1 + cot²θ = csc²θ — derived by dividing the first identity by sin²θ
- tan²θ + 1 = sec²θ — derived by dividing by cos²θ
That last point matters. You don't need three separate memories. If you forget tan²θ + 1 = sec²θ, just divide sin²θ + cos²θ = 1 by cos²θ and you get it instantly.
Reciprocal Identities
These define the relationships between the six trig functions. Simple division.
- csc θ = 1/sin θ
- sec θ = 1/cos θ
- cot θ = 1/tan θ = cos θ/sin θ
That's it. Three definitions. The other three (sin, cos, tan) are your primary functions.
Quotient Identities
Tan and cot are defined as ratios of sin and cos:
- tan θ = sin θ/cos θ
- cot θ = cos θ/sin θ
These aren't new information. They're just definitions. But they let you rewrite tangent in terms of sine and cosine, which becomes useful when you need to simplify expressions.
Cofunction Identities
These describe what happens when you work with 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
The same pattern holds for cosecant and secant. The cofunction of an angle equals the function of its complement. That's the whole rule.
Even and Odd Properties
These tell you how trig functions behave with negative angles:
- cos(-θ) = cos θ — cosine is even
- sin(-θ) = -sin θ — sine is odd
- tan(-θ) = -tan θ — tangent is odd
Secant inherits cosine's even property. Cosecant and cotangent inherit odd properties from their counterparts.
Sum and Difference Formulas
These let you break apart angles or combine them. You'll use these constantly in calculus.
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)
Notice the pattern: the sign in the middle changes between formulas. Memorize one and adjust the signs for the other.
Double Angle Formulas
These are special cases of the sum formulas where A = B.
- sin(2θ) = 2 sin θ cos θ
- cos(2θ) = cos²θ - sin²θ = 2cos²θ - 1 = 1 - 2sin²θ
- tan(2θ) = 2 tan θ / (1 - tan²θ)
The cosine formula has three forms. Pick whichever matches what you know. If you know sin²θ, use 1 - 2sin²θ. If you know cos²θ, use 2cos²θ - 1.
Half Angle Formulas
These come from solving the double angle formulas for the original angle. They're messier.
- sin(θ/2) = ±√((1 - cos θ)/2)
- cos(θ/2) = ±√((1 + cos θ)/2)
- tan(θ/2) = ±√((1 - cos θ)/(1 + cos θ)) = sin θ/(1 + cos θ) = (1 - cos θ)/sin θ
The ± sign depends on which quadrant the half angle falls in. That's the part most students forget to check.
Product-to-Sum Formulas
These convert products into sums. Useful for integration later.
- 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)]
The reverse (sum-to-product) exists too, but you can derive those by reversing the process.
Quick Reference Table
| Identity Type | Key Formula |
|---|---|
| Pythagorean | sin²θ + cos²θ = 1 |
| Pythagorean | 1 + tan²θ = sec²θ |
| Pythagorean | 1 + cot²θ = csc²θ |
| Reciprocal | csc θ = 1/sin θ |
| Reciprocal | sec θ = 1/cos θ |
| Quotient | tan θ = sin θ/cos θ |
| Cofunction | sin(π/2 - θ) = cos θ |
| Double Angle | sin(2θ) = 2 sin θ cos θ |
| Double Angle | cos(2θ) = cos²θ - sin²θ |
How to Actually Use These
Knowing the formulas isn't enough. You need to know when to use them.
Simplifying Expressions
When you see a messy trig expression, convert everything to sin and cos. Then look for opportunities to cancel, combine, or apply Pythagorean identities.
Example: simplify sec θ - cos θ
Convert: 1/cos θ - cos θ
Common denominator: (1 - cos²θ)/cos θ
Pythagorean identity: sin²θ/cos θ = sin θ tan θ
Proving Identities
There's no single right path. But these moves work:
- Convert everything to sin and cos
- Multiply numerator and denominator by the same expression (usually the conjugate)
- Factor and cancel common factors
- Use Pythagorean identities when you see sin² or cos²
Start with the messier side. It's usually easier to build toward a simpler expression than to break down a simple one.
Solving Equations
Trig equations aren't identities. They only need to hold for specific values. Here's the process:
- Isolate the trig function
- Find the reference angle
- Determine all solutions within the given interval
- Check for extraneous solutions (especially when you divide by something)
What to Actually Memorize
You don't need to memorize everything above. Memorize the core set and derive the rest:
- sin²θ + cos²θ = 1
- sin(A ± B) and cos(A ± B)
- The reciprocal definitions
From sin²θ + cos²θ = 1, you get the other Pythagorean identities. From the sum formulas, you get double angle. From double angle, you get half angle.
Everything connects back to the Pythagorean theorem and the unit circle. That's the foundation. Build from there.