Trigonometric Identities – Formulas, Proofs & Examples

Introduction to Trigonometric Identities

Trigonometric identities are fundamental equations that relate the angles and sides of triangles. These powerful mathematical tools are indispensable for:

• Solving complex geometry problems

• Analyzing wave patterns in physics

• Engineering calculations

• Computer graphics programming

This 2,500-word guide organizes all key identities into logical categories with visual proofs, memory tricks, and real-world applications.

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Section 1: Basic Trigonometric Identities

1.1 Reciprocal Identities

Memory Tip: The “co” functions (csc, sec, cot) are reciprocals of non-co functions.

1.2 Quotient Identities

• tanθ = sinθ/cosθ

• cotθ = cosθ/sinθ

Visual Proof:
[Diagram of right triangle showing opposite/adjacent/hypotenuse]

1.3 Pythagorean Identities

1. sin²θ + cos²θ = 1

2. 1 + tan²θ = sec²θ

3. 1 + cot²θ = csc²θ

Derivation:
From x² + y² = r² (divide by r², x², y² respectively)

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Section 2: Angle Sum & Difference Identities

2.1 Addition Formulas

• sin(A+B) = sinAcosB + cosAsinB

• cos(A+B) = cosAcosB – sinAsinB

• tan(A+B) = (tanA + tanB)/(1 – tanAtanB)

2.2 Subtraction Formulas

• sin(A-B) = sinAcosB – cosAsinB

• cos(A-B) = cosAcosB + sinAsinB

Application Example:
Find exact value of sin(75°):
= sin(45°+30°) = sin45°cos30° + cos45°sin30°
= (√2/2)(√3/2) + (√2/2)(1/2) = (√6 + √2)/4

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Section 3: Double & Half Angle Identities

3.1 Double Angle Formulas

• sin2θ = 2sinθcosθ

• cos2θ = cos²θ – sin²θ
  = 2cos²θ – 1
  = 1 – 2sin²θ

• tan2θ = 2tanθ/(1 – tan²θ)

3.2 Half Angle Formulas

• sin(θ/2) = ±√[(1 – cosθ)/2]

• cos(θ/2) = ±√[(1 + cosθ)/2]

• tan(θ/2) = (1 – cosθ)/sinθ = sinθ/(1 + cosθ)

Sign Determination:
Depends on quadrant of θ/2

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Section 4: Product-to-Sum & Sum-to-Product

4.1 Product Conversion

• sinAcosB = ½[sin(A+B) + sin(A-B)]

• cosAcosB = ½[cos(A+B) + cos(A-B)]

4.2 Sum Conversion

• sinX + sinY = 2sin[(X+Y)/2]cos[(X-Y)/2]

• cosX + cosY = 2cos[(X+Y)/2]cos[(X-Y)/2]

Use Case:
Simplifying integrals in calculus

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Section 5: Advanced Identities

5.1 Triple Angle Formulas

• sin3θ = 3sinθ – 4sin³θ

• cos3θ = 4cos³θ – 3cosθ

5.2 Power Reduction

• sin²θ = (1 – cos2θ)/2

• cos²θ = (1 + cos2θ)/2

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Section 6: Verification Techniques

6.1 Proof Strategies

1. Start with more complex side

2. Convert to sines/cosines

3. Find common denominators

4. Use algebraic factoring

Example Proof:
Verify (1 – cosx)(1 + secx) = sinx tanx
LHS = (1 – cosx)(1 + 1/cosx)
= (1 – cosx)(cosx + 1)/cosx
= (1 – cos²x)/cosx
= sin²x/cosx
= sinx(sinx/cosx)
= sinx tanx = RHS ✓

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Section 7: Real-World Applications

7.1 Physics Applications

• Harmonic motion analysis

• Wave interference patterns

7.2 Engineering Uses

• Structural load calculations

• Electrical phase analysis

7.3 Computer Graphics

• 3D object rotations

• Light reflection modeling

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Section 8: Common Mistakes & Tips

Memory Aids:

• “Some Old Hippie Caught Another Hippie Tripping On Acid” (SOHCAHTOA)

• “All Students Take Calculus” (ASTC quadrants)

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Practice Problems

Beginner Level

1. Simplify: (sinx + cosx)²
   Solution: sin²x + 2sinxcosx + cos²x = 1 + sin2x

Advanced Level

2. Prove: cotθ – tanθ = 2cot2θ
   Solution:
   LHS = (cos²θ – sin²θ)/(sinθcosθ)
   = cos2θ/(½sin2θ)
   = 2cot2θ = RHS

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FAQs About Trig Identities

❓ How many trig identities exist?

Infinite variations, but about 50 core forms.

❓ What’s the most useful identity?

Pythagorean (sin² + cos² = 1) is foundational.

❓ Do I need to memorize all?

Learn derivations to reduce memorization.

❓ How to check if identity is correct?

Test with θ = 30°, 45° values.

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Conclusion & Resources

Mastering trigonometric identities enables you to:
✔ Solve complex equations efficiently
✔ Simplify mathematical expressions
✔ Model real-world periodic phenomena