- tan θ cos θ = sin(-θ)
- Start with left side: -tan θ cos θ
- Substitute tan θ = sin θ / cos θ: -(sin θ / cos θ) * cos θ
- Simplify: -sin θ
- Use odd function property: sin(-θ) = -sin θ
- Therefore, -tan θ cos θ = sin(-θ)
cot² θ (1 + tan² θ) = csc² θ
- Start with left side: cot² θ (1 + tan² θ)
- Use Pythagorean identity: 1 + tan² θ = sec² θ
- Substitute: cot² θ * sec² θ
- Express in terms of sin and cos: (cos² θ / sin² θ) * (1 / cos² θ)
- Simplify: 1 / sin² θ
- Recognize: csc² θ = 1 / sin² θ
- Therefore, cot² θ (1 + tan² θ) = csc² θ
sec θ / csc θ = tan θ
- Start with left side: sec θ / csc θ
- Express in terms of sin and cos: (1 / cos θ) / (1 / sin θ)
- Simplify division: (1 / cos θ) * (sin θ / 1) = sin θ / cos θ
- Recognize: tan θ = sin θ / cos θ
- Therefore, sec θ / csc θ = tan θ
sin² θ (csc² θ - 1) = cos² θ
- Start with left side: sin² θ (csc² θ - 1)
- Use Pythagorean identity: csc² θ - 1 = cot² θ
- Substitute: sin² θ * cot² θ
- Express cot² θ: sin² θ * (cos² θ / sin² θ)
- Simplify: cos² θ
- Therefore, sin² θ (csc² θ - 1) = cos² θ
cot θ sin θ = cos θ
- Start with left side: cot θ sin θ
- Express cot θ: (cos θ / sin θ) * sin θ
- Simplify: cos θ
- Therefore, cot θ sin θ = cos θ
(sec θ - 1)(sec θ + 1) = tan² θ
- Start with left side: (sec θ - 1)(sec θ + 1)
- Apply difference of squares: sec² θ - 1
- Use Pythagorean identity: sec² θ - 1 = tan² θ
- Therefore, (sec θ - 1)(sec θ + 1) = tan² θ
sec θ cot θ sin θ = 1
- Start with left side: sec θ cot θ sin θ
- Express each function: (1 / cos θ) * (cos θ / sin θ) * sin θ
- Simplify: (1 / cos θ) * cos θ = 1
- Therefore, sec θ cot θ sin θ = 1
(1 - cos θ)(1 + sec θ) = sec θ - cos θ
- Start with left side: (1 - cos θ)(1 + sec θ)
- Expand using distributive property: 1 * 1 + 1 * sec θ - cos θ * 1 - cos θ * sec θ
- Simplify: 1 + sec θ - cos θ - (cos θ * 1 / cos θ)
- Simplify further: 1 + sec θ - cos θ - 1
- Combine like terms: sec θ - cos θ
- Therefore, (1 - cos θ)(1 + sec θ) = sec θ - cos θ
cos θ csc θ = cot θ
- Start with left side: cos θ csc θ
- Express csc θ: cos θ * (1 / sin θ)
- Simplify: cos θ / sin θ
- Recognize: cot θ = cos θ / sin θ
- Therefore, cos θ csc θ = cot θ
(cos θ + sin θ) / sin θ = 1 + cot θ
- Start with left side: (cos θ + sin θ) / sin θ
- Split fraction: cos θ / sin θ + sin θ / sin θ
- Simplify: cot θ + 1
- Rearrange: 1 + cot θ
- Therefore, (cos θ + sin θ) / sin θ = 1 + cot θ
Parent Tip: Review the logic above to help your child master the concept of verifying trigonometric identities worksheet.
