PanMaths.comFree Printable Math Formula Reference Sheet
Trigonometric Identities
Trigonometry • Grades 10, 11, 12
Trigonometry
Trigonometric Identities
Essential identities for simplifying trig expressions.
The Formula
sin²θ + cos²θ = 1
Used for: Simplifying complex trigonometric expressions, derivatives, integrals, and proofs
sin θSine of angle θ (opposite / hypotenuse)
cos θCosine of angle θ (adjacent / hypotenuse)
tan θTangent of angle θ (sin θ / cos θ)
⚡Live Interactive Quick Solver
Instant Calculationdegrees
Pythagorean Identity Check:sin²(45°) + cos²(45°) = 1.0000 ≡ 1 ✓
• sin(45°) = 0.7071 ➔ sin² = 0.5000• cos(45°) = 0.7071 ➔ cos² = 0.5000• Sum: 0.5000 + 0.5000 = 1.0000
\sin^2(\theta) + \cos^2(\theta) = 1
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📋 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!
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