Trigonometric identities are equalities that involve trigonometric functions and are true for every value of the occurring variables for which both sides of the equality are defined. There are many trigonometric identities, but some of the most important ones are:
1. Reciprocal identities:
- sin(θ) = 1/csc(θ)
- cos(θ) = 1/sec(θ)
- tan(θ) = 1/cot(θ)
2. Pythagorean identities:
- sin^2(θ) + cos^2(θ) = 1
- 1 + tan^2(θ) = sec^2(θ)
- 1 + cot^2(θ) = csc^2(θ)
3. Quotient identities:
- tan(θ) = sin(θ)/cos(θ)
- cot(θ) = cos(θ)/sin(θ)
4. Co-function identities:
- sin(π/2 - θ) = cos(θ)
- cos(π/2 - θ) = sin(θ)
- tan(π/2 - θ) = cot(θ)
- cot(π/2 - θ) = tan(θ)
5. Even-odd identities:
- sin(-θ) = -sin(θ)
- cos(-θ) = cos(θ)
- tan(-θ) = -tan(θ)
6. Sum and difference identities:
- sin(α + β) = sin(α)cos(β) + cos(α)sin(β)
- cos(α + β) = cos(α)cos(β) - sin(α)sin(β)
- tan(α + β) = (tan(α) + tan(β))/(1 - tan(α)tan(β))
7. Double angle identities:
- sin(2θ) = 2sin(θ)cos(θ)
- cos(2θ) = cos^2(θ) - sin^2(θ)
- tan(2θ) = (2tan(θ))/(1 - tan^2(θ))
8. Half angle identities:
- sin(θ/2) = ±√[(1 - cos(θ))/2]
- cos(θ/2) = ±√[(1 + cos(θ))/2]
- tan(θ/2) = ±√[(1 - cos(θ))/(1 + cos(θ))]
These are just some of the most commonly used trigonometric identities. There are many more identities that can be derived from these basic ones.