Advanced Functions: Proving Trigonometric Identities

Introduction

• Proving trigonometric identities can be challenging but becomes easier with practice.
• The process becomes a puzzle-solving exercise.
• This lesson introduces the basics and some initial examples, with a follow-up lesson planned for more complex identities.

General Rules for Proving Identities

1. Identify the more complicated side:
   • Generally start simplifying the more complex side.
2. Work each side separately:
   • Avoid crossing the equal sign; simplify each side independently.
3. Break down into sines and cosines:
   • This helps see similarities and simplify expressions.
4. Find a common denominator:
   • Especially when you have two terms on one side and one on the other.
5. Use factoring techniques:
   • Apply difference of cubes, difference of squares as needed.
6. Use substitutions:
   • Utilize known identities like (\sin^2(A) + \cos^2(A) = 1).
7. Be Creative and Persistent:
   • Use conjugates and other techniques as required. Practice enhances skill.

Example Problems

Example 1:

• Identity: ( \sin x \tan x = \sec x - \cos x )
• Solution Steps:
  • Replace ( \tan x = \frac{\sin x}{\cos x} ).
  • Simplify left side: ( \sin^2 x / \cos x ).
  • Simplify right side: ( 1 / \cos x - \cos x ).
  • Find common denominator: ( \cos x ).
  • Recognize ( 1 - \cos^2 x = \sin^2 x ).
  • Conclusion: Left side equals right side (QED).

Example 2:

• Identity: ( 1 + \cot x \tan y = \sin(x+y) )
• Solution Steps:
  • Break down ( \cot x = \frac{\cos x}{\sin x} ) and ( \tan y = \frac{\sin y}{\cos y} ).
  • Simplify left side combining terms: ( 1 + \frac{\cos x \sin y}{\sin x \cos y} ).
  • Simplify right side using angle addition: ( \sin x \cos y + \cos x \sin y ).
  • Show equivalence of both sides by common denominator.

Example 3:

• Identity: ( \cos^2 \theta - \sin^2 \theta = 1 - \tan \theta )
• Solution Steps:
  • Recognize ( \cos^2 \theta - \sin^2 \theta ) as a difference of squares.
  • Factor and simplify: ( (\cos \theta + \sin \theta)(\cos \theta - \sin \theta) ).
  • Simplify using known identities to show equivalence to ( 1 - \tan \theta ).

Example 4:

• Identity: ( \sin(2x)(1 - \cos(2x)) = 2\csc(2x) - \tan x )
• Solution Steps:
  • Simplify ( \sin(2x) = 2\sin x \cos x ).
  • Replace ( \cos(2x) ) with known identity forms.
  • Simplify to match both sides using common denominators and substitution.

Conclusion

• Practice is essential to become proficient in proving identities.
• For further assistance or specific problems, reach out for help.
• Next lesson will cover more complex examples.

Note: Use QED (Quod Erat Demonstrandum) to indicate the end of a proof.