Verifying Trigonometric Identities Lecture Notes

Introduction

• Goal: Solve 14 trigonometric identity verification problems.
• Approach: Start with the more complex side of the equation and simplify to match the simpler side.

Problem 1: Prove ( \frac{\csc^2\theta - 1}{\csc^2\theta} = \cos^2\theta )

1. Simplify the left side: ( \csc^2\theta - 1 = \cot^2\theta ).
2. Replace ( \cot^2\theta = \frac{\cos^2\theta}{\sin^2\theta} ).
3. Replace ( \csc^2\theta = \frac{1}{\sin^2\theta} ).
4. Cancel ( \sin^2\theta ) to obtain ( \cos^2\theta ).

Problem 2: Simplify and combine fractions

1. Identify the need for a common denominator.
2. Multiply fractions to obtain a common denominator.
3. Simplify using the Pythagorean identity: ( 1 - \cos^2\theta = \sin^2\theta ).
4. Result: ( 2 \csc^2\theta ).

Problem 3: Prove ( \cot^2\theta + 1 = \csc^2\theta )

1. Replace given trigonometric identities.
2. Simplify using ( \csc^2 - 1 = \cot^2 ).
3. Result: (-\cot^2\theta ).

Problem 4: Use conjugates for simplification

1. Multiply by conjugate to simplify expressions.
2. Expand and simplify: ( \sin^2\theta = 1 - \cos^2\theta ).
3. Result: ( \csc\theta + \cot\theta ).

Problem 5: Factor common terms

1. Factor out ( \cot^2\theta ) in expressions.
2. Simplify using Pythagorean identities.
3. Result: ( \cot^4\theta ).

Problem 6: Combine into one fraction

1. Find common denominator for terms.
2. Simplify using trig identities: ( 2 \csc\theta ).

Problem 7: Use cofunction identities

1. Use ( \cot(\frac{\pi}{2} - \theta) = \tan\theta ).
2. Simplify: ( \tan\theta \cdot \cot\theta = 1 ).

Problem 8: Even and odd identities

1. Apply identities to negative angles.
2. Simplify: ( -\tan\theta ).

Problem 9: Factorization of complex expressions

1. Factor out common terms ( \csc^2\theta \cot\theta ).
2. Simplify using Pythagorean identity.
3. Result: ( \csc^3\theta \cot^3\theta ).

Problem 10: FOIL and simplify

1. FOIL out expressions.
2. Use ( \sin^2\theta = 1 - \cos^2\theta ).

Problem 11: Common denominators and complex fractions

1. Simplify into one expression.
2. Use identities to reduce: ( \sec\theta ).

Problem 12: Co-function identities

1. Apply ( \tan(\frac{\pi}{2} - \theta) = \cot\theta ).
2. Simplify: ( \csc^2 - \cot^2 = 1 ).

Problem 13: Combine into one fraction

1. Use common denominator to combine terms.
2. Simplify using identities: (-2 \tan\theta \sec\theta ).

Problem 14: Start with left side

1. Simplify: ( \tan\theta = \frac{\sin\theta}{\cos\theta} ).
2. Simplify using identities: ( \sec\theta - \cos\theta ).

Conclusion

• Practice and familiarity with identities are key to mastering these problems.
• Review past problems and techniques for a deeper understanding.