How to Prove Trigonometric Identities (Precalculus - Trigonometry 24)

Overview

Professor Leonard outlines a systematic approach to proving trigonometric identities, emphasizing working on the "harder" side, simplifying fractions, and converting to sines and cosines when necessary. He demonstrates this methodology through 11 examples, covering techniques like factoring, using Pythagorean and reciprocal identities, and manipulating expressions to match the target side, highlighting the importance of critical thinking and recognizing patterns.

Key takeaways

To prove a trigonometric identity, start with the more complex side and manipulate it using known identities and algebraic rules.
When stuck, converting all terms to sines and cosines is a reliable strategy, but not always the most efficient.
Look for opportunities to use Pythagorean identities (e.g., sin²(θ) + cos²(θ) = 1) and reciprocal identities (e.g., sec(θ) = 1/cos(θ)) as soon as they become apparent.
Factoring, especially to create differences of squares, can be crucial for simplifying complex expressions.
Even/odd identities (e.g., sin(-θ) = -sin(θ), cos(-θ) = cos(θ)) are essential when dealing with negative angles.
Sometimes, strategic multiplication by a term over itself (similar to multiplying by a conjugate) is needed to create a desired structure, like a difference of squares in the denominator.

Chapters

0:00 Introduction to Proving Trigonometric Identities

Identities are equations that hold true for all valid inputs.
The general strategy involves working on the more complex side of the identity.
Key steps include simplifying, writing in terms of sine and cosine, and using known identities.

1:01 General Strategy for Proving Identities

Identify the 'harder' side (typically the one with more terms or complexity).
Work only on that side to transform it into the 'easier' side.
Prioritize simplifying fractions, applying apparent identities, and converting to sines and cosines.

5:36 Example 1: Proving sec(θ)sin(θ) = tan(θ)

Start with the harder side: sec(θ)sin(θ).
Rewrite sec(θ) as 1/cos(θ).
Combine terms to get sin(θ)/cos(θ), which simplifies to tan(θ).

10:00 Example 2: Proving (sec(θ) - 1)/(sec(θ) + 1) = tan²(θ)

Begin with the left side: (sec(θ) - 1)/(sec(θ) + 1).
Distribute to get sec²(θ) - 1 in the numerator.
Use the identity sec²(θ) - 1 = tan²(θ) to simplify.

13:48 Example 3: Proving cos(θ)tan(θ) + cot(θ) = csc(θ)

Start with cos(θ)tan(θ) + cot(θ).
Convert tan(θ) to sin(θ)/cos(θ) and cot(θ) to cos(θ)/sin(θ).
Simplify cos(θ) * (sin(θ)/cos(θ)) to sin(θ).
Combine sin(θ) + cos²(θ)/sin(θ) into a single fraction (sin²(θ) + cos²(θ))/sin(θ), which equals 1/sin(θ) or csc(θ).

23:43 Example 4: Proving sin(θ)csc(θ) - cos²(θ) = sin²(θ)

Start with sin(θ)csc(θ) - cos²(θ).
Substitute csc(θ) with 1/sin(θ) to get sin(θ) * (1/sin(θ)) - cos²(θ).
Simplify to 1 - cos²(θ), which is the Pythagorean identity for sin²(θ).

26:53 Example 5: Proving (1 - cos²(θ))(1 + cot²(θ)) = 1

Recognize 1 - cos²(θ) as sin²(θ) and 1 + cot²(θ) as csc²(θ).
Substitute these identities: sin²(θ) * csc²(θ).
Rewrite csc²(θ) as 1/sin²(θ) to get sin²(θ) * (1/sin²(θ)), which simplifies to 1.

31:40 Example 6: Proving sec⁴(θ) - sec²(θ) = tan⁴(θ) + tan²(θ)

Factor sec²(θ) from the left side: sec²(θ)(sec²(θ) - 1).
Use the identity sec²(θ) - 1 = tan²(θ) to get sec²(θ)tan²(θ).
Substitute sec²(θ) with tan²(θ) + 1: (tan²(θ) + 1)tan²(θ).
Distribute to get tan⁴(θ) + tan²(θ).

38:47 Example 7: Proving 3sin²(θ) + 4cos²(θ) = 3 + cos²(θ)

Rewrite 4cos²(θ) as 3cos²(θ) + cos²(θ) on the left side.
Group terms: 3sin²(θ) + 3cos²(θ) + cos²(θ).
Factor out 3: 3(sin²(θ) + cos²(θ)) + cos²(θ).
Use the Pythagorean identity sin²(θ) + cos²(θ) = 1 to get 3(1) + cos²(θ) = 3 + cos²(θ).

43:29 Example 8: Proving (sin²(-θ) - cos²(-θ))/(sin(-θ) - cos(-θ)) = cos(θ) - sin(θ)

Use even/odd identities: sin(-θ) = -sin(θ) and cos(-θ) = cos(θ).
Substitute into the expression: ((-sin(θ))² - cos²(θ))/(-sin(θ) - cos(θ)).
Simplify numerator to sin²(θ) - cos²(θ).
Factor numerator as a difference of squares: (sin(θ) - cos(θ))(sin(θ) + cos(θ)).
The expression becomes (sin(θ) - cos(θ))(sin(θ) + cos(θ))/(-sin(θ) - cos(θ)).
Factor out -1 from the denominator: (sin(θ) - cos(θ))(sin(θ) + cos(θ))/-(sin(θ) + cos(θ)).
Cancel (sin(θ) + cos(θ)) to get -(sin(θ) - cos(θ)), which is cos(θ) - sin(θ).

54:16 Example 9: Proving (1 - cot²(θ))/(1 + cot²(θ)) + 2csc(θ)cos²(θ) = 1

Substitute 1 + cot²(θ) with csc²(θ) in the first term.
Rewrite the first term as (1 - cot²(θ))/csc²(θ).
Convert cot²(θ) to cos²(θ)/sin²(θ) and csc²(θ) to 1/sin²(θ).
Simplify the first term to (sin²(θ) - cos²(θ))/sin²(θ) * sin²(θ) = sin²(θ) - cos²(θ).
Rewrite the second term: 2 * (1/sin(θ)) * cos²(θ) = 2cos²(θ)/sin(θ).
Combine terms and simplify to get 1.

Summary, takeaways, and chapters were generated by AI from the video's transcript and may contain errors. The video belongs to its creator, Professor Leonard.