Lecture Notes: Verifying Trigonometric Identities

Introduction

• Focus on 14 problems of verifying trigonometric identities.
• Approach: Determine which side of the identity is more complex and simplify it.
• Key tool: Pythagorean trigonometric identities.

Problem 1

• Identity: ( \frac{\csc^2\theta - 1}{\csc^2\theta} = \cos^2\theta )
• Steps:
  1. Replace ( \csc^2\theta - 1 ) with ( \cot^2\theta ).
  2. Convert ( \cot^2 ) to ( \frac{\cos^2\theta}{\sin^2\theta} ).
  3. Replace ( \csc^2\theta ) with ( \frac{1}{\sin^2\theta} ).
  4. Simplify to ( \cos^2\theta ).

Problem 2

• Identity: Simplify two fractions into one.
• Steps:
  1. Get a common denominator for both fractions.
  2. Use Pythagorean identities to simplify.
  3. Result: ( 2\csc^2\theta ).

Problem 3

• Identity: ( \cot^2\theta + 1 )(( \sin^2\theta - 1 )) = (-\cot^2\theta)
• Steps:
  1. Use Pythagorean identities to simplify each part.
  2. Convert to basic trigonometric functions.
  3. Simplify to match the right side.

Problem 4

• Identity: ( \csc\theta + \cot\theta = \frac{\sin\theta}{1-\cos\theta} )
• Steps:
  1. Multiply by the conjugate of the denominator.
  2. Simplify using identities and cancellations.

Problem 5

• Identity: ( \cot^4\theta = \cot^2\theta \csc^2\theta - \cot^2\theta )
• Steps:
  1. Factor out common terms.
  2. Simplify using Pythagorean identities.

Problem 6

• Identity: ( \frac{1+\cos\theta}{\sin\theta} + \frac{\sin\theta}{1+\cos\theta} = 2\csc\theta )
• Steps:
  1. Combine fractions with a common denominator.
  2. Simplify using trigonometric identities.

Problem 7 & 8

• 7: Use cofunction identities to simplify.
• 8: Use even and odd identities to verify ( -\tan\theta ).

Problem 9

• Identity: Complex expression equals ( \csc^3\theta \cot^3\theta )
• Steps:
  1. Factor out greatest common factors.
  2. Simplify using identities.

Problem 10

• Identity: Simplify ( 1+\cos\theta ) form.
• Steps:
  1. Use even and odd identities.
  2. Simplify the binomial product.

Problem 11

• Identity: ( \frac{\sin\theta \tan\theta}{1-\cos\theta} - 1 = \sec\theta )
• Steps:
  1. Combine terms over a common denominator.
  2. Simplify and factor.

Problem 12

• Identity: ( \csc^2\theta - \tan^2(\frac{\pi}{2} - \theta) = 1 )
• Steps:
  1. Convert tan to cot using cofunction identities.
  2. Use Pythagorean identities.

Problem 13

• Identity: Combine two fraction expressions.
• Steps:
  1. Obtain a common denominator.
  2. Simplify using Pythagorean identities.

Problem 14

• Identity: ( \frac{\tan\theta}{\csc\theta} = \sec\theta - \cos\theta )
• Steps:
  1. Convert all to sine and cosine.
  2. Simplify using basic identities.

Conclusion

• Practice makes perfect in recognizing and applying identities.
• Each identity requires a unique approach but relies on foundational identities.