Trigonometry
Trigonometric Identities
Essential identities for simplifying trig expressions.
The Formula
sin²θ + cos²θ = 1
Used for: Simplifying trigonometric expressions and proving equations
sin θSine of angle θ
cos θCosine of angle θ
tan θTangent (sin/cos)
📋 Key Identities
Organized by type:
📐
Pythagoreansin²θ + cos²θ = 1
1 + tan²θ = sec²θ
1 + cot²θ = csc²θ
📐
Quotienttan θ = sin θ / cos θ
cot θ = cos θ / sin θ
📐
Reciprocalcsc θ = 1/sin θ
sec θ = 1/cos θ
cot θ = 1/tan θ
📝 Worked Examples
Simplify Using Pythagorean
Problem:
Simplify: sin²θ + cos²θ + tan²θ1
Use identity
sin²θ + cos²θ = 12
Substitute
1 + tan²θ3
Use identity
1 + tan²θ = sec²θAnswer: sec²θ
Prove Identity
Problem:
Prove: tanθ·cosθ = sinθ1
Substitute tanθ
(sinθ/cosθ)·cosθ2
Cancel cosθ
sinθAnswer: Proven! ✓
💡 Pro Tips
Start with one side
When proving, work on ONE side of the equation only
Convert to sin/cos
When stuck, convert everything to sin and cos
Memorize Pythagorean
sin²θ + cos²θ = 1 is the most used identity!