Lecture Notes: Verifying Trigonometric Identities

Key Concepts

• Objective: Show that the left side of a trigonometric equation is equivalent to the right side.
• Techniques: Six techniques to verify identities
  1. Convert to Sine and Cosine: A good start for easier problems.
  2. Combine Terms: If left side has two terms and right side has one, combine terms using common denominators or factoring.
  3. Split Terms: If left side has one term and right side has two, split terms by distributing or splitting fractions.
  4. Convert Division to Multiplication: Use reciprocal identities and algebraic manipulation.
  5. Multiply by Conjugate: Useful when fractions have sums or differences involving ± signs.
  6. Factor or Foil: Use algebraic techniques to factor or distribute terms.

Important Trigonometric Identities

• Pythagorean Identities:

  • ( \sin^2(x) + \cos^2(x) = 1 )
  • ( 1 + \tan^2(x) = \sec^2(x) )
  • ( 1 + \cot^2(x) = \csc^2(x) )

• Reciprocal Identities:

  • ( \tan(x) = \frac{\sin(x)}{\cos(x)} )
  • ( \cot(x) = \frac{\cos(x)}{\sin(x)} )
  • ( \sec(x) = \frac{1}{\cos(x)} )
  • ( \csc(x) = \frac{1}{\sin(x)} )

Practice Problems

Example 1: Verify ( \sin(x) \cdot \sec(x) = \tan(x) )

• Solution: Convert ( \sec(x) ) to ( \frac{1}{\cos(x)} ), verify ( \frac{\sin(x)}{\cos(x)} = \tan(x) ).

Example 2: ( \tan^2(x) \cdot \cot^2(x) = 1 )

• Solution: Convert ( \tan(x) ) and ( \cot(x) ) to sine and cosine, simplify to show it equals 1.

Example 3: ( \cot(x) \cdot \sec(x) \cdot \sin(x) = 1 )

• Solution: Convert to sine and cosine, simplify using cancellations.

Example 4: ( \frac{\cos(x) \cdot \sec(x)}{\cot(x)} = \tan(x) )

• Solution: Simplify numerator using ( \sec(x) = \frac{1}{\cos(x)} ), use ( \frac{1}{\cot(x)} = \tan(x) ).

Example 5: ( \sin(x) \cdot \tan(x) = \frac{1 - \cos^2(x)}{\cos(x)} )

• Solution: Convert ( \tan(x) ) to sine and cosine, use ( \sin^2(x) = 1 - \cos^2(x) ).

Example 6: ( \cos^2(x) - \sin^2(x) = 1 - 2\sin^2(x) )

• Solution: Replace ( \cos^2(x) ) with ( 1 - \sin^2(x) ) using identities.

Example 7: ( \sin(x) \cdot \tan(x) + \cos(x) = \sec(x) )

• Solution: Convert left side to common denominator, use ( \sin^2(x) + \cos^2(x) = 1 ).

Example 8: ( \sec(x) - \cos(x) = \tan(x) \cdot \sin(x) )

• Solution: Convert each term to sine and cosine, simplify using common denominators and identities.

Problem Solving Tips

• Use Right Side as a Guide: Helps to know what transformations or identities to apply.
• Common Denominators: Useful for combining terms into a single fraction.
• Identities: Essential to simplify and verify equations.