Basic Trigonometric Identities

Pythagorean Identities (and their variations)

From the unit circle, using the Pythagorean Theorem, we see that

x² + y² = 1

Since x = cos θ & y = sin θ, we get:

(sin θ)² + (cos θ)² = 1 or

sin² θ + cos² θ = 1

1. sin2 θ + cos2 θ = 1 (sin2 θ = 1 - cos2 θ) (cos2 θ = 1 - sin2 θ)

2. tan2 θ + 1 = sec2 θ (tan2 θ - sec2 θ = 1) (tan2 θ = sec2 θ – 1)

3. 1 + cot2 θ = csc2 θ (cot2 θ =  csc2 θ – 1) (1 =  csc2 θ - cot2 θ)

Opposite Angle Identities (aka Even/Odd Identities)

1. sin (-A) = - sin (A)                2. cos (-A) = cos (A)

Sum and Difference Identities

1.    sin(A + B) = sin(A)cos(B) + sin(B)cos(A)

2.  sin(A - B) = sin(A)cos(B) - sin(B)cos(A)

3.  cos(A + B) = cos(A)cos(B) – sin(A)sin(B)

4.  cos(A – B) = cos(A)cos(B) + sin(A)sin(B)

Cofunction Identities:  

1. sin θ = cos (90° – θ)	4. cos θ = sin (90° – θ)
2. tan θ = cot (90° – θ)	5. cot θ = tan (90° – θ)
3. sec θ = csc (90° – θ)	6. csc θ = sec (90° – θ)