- tan θ cos θ = sin(-θ)
- Left side: -tan θ cos θ = -(sin θ / cos θ) * cos θ = -sin θ
- Right side: sin(-θ) = -sin θ (sine is odd)
- Both sides equal -sin θ → Verified
cot² θ (1 + tan² θ) = csc² θ
- Left side: cot² θ (1 + tan² θ) = cot² θ * sec² θ (since 1 + tan² θ = sec² θ)
- = (cos² θ / sin² θ) * (1 / cos² θ) = 1 / sin² θ = csc² θ
- Equals right side → Verified
sec θ / csc θ = tan θ
- Left side: sec θ / csc θ = (1 / cos θ) / (1 / sin θ) = sin θ / cos θ = tan θ
- Equals right side → Verified
sin² θ (csc² θ - 1) = cos² θ
- Left side: sin² θ (csc² θ - 1) = sin² θ (1/sin² θ - 1) = sin² θ * (1 - sin² θ)/sin² θ = 1 - sin² θ = cos² θ
- Equals right side → Verified
cot θ sin θ = cos θ
- Left side: cot θ sin θ = (cos θ / sin θ) * sin θ = cos θ
- Equals right side → Verified
(sec θ - 1)(sec θ + 1) = tan² θ
- Left side: (sec θ - 1)(sec θ + 1) = sec² θ - 1
- = tan² θ (Pythagorean identity)
- Equals right side → Verified
sec θ cot θ sin θ = 1
- Left side: sec θ cot θ sin θ = (1 / cos θ) * (cos θ / sin θ) * sin θ = 1
- Equals right side → Verified
(1 - cos θ)(1 + sec θ) = sec θ - cos θ
- Left side: (1 - cos θ)(1 + sec θ) = (1 - cos θ)(1 + 1/cos θ) = (1 - cos θ)((cos θ + 1)/cos θ)
- = (1 - cos θ)(1 + cos θ)/cos θ = (1 - cos² θ)/cos θ = sin² θ / cos θ
- Right side: sec θ - cos θ = 1/cos θ - cos θ = (1 - cos² θ)/cos θ = sin² θ / cos θ
- Both sides equal → Verified
cos θ csc θ = cot θ
- Left side: cos θ csc θ = cos θ * (1 / sin θ) = cos θ / sin θ = cot θ
- Equals right side → Verified
(cos θ + sin θ) / sin θ = 1 + cot θ
- Left side: (cos θ + sin θ) / sin θ = cos θ / sin θ + sin θ / sin θ = cot θ + 1
- Equals right side → Verified
Parent Tip: Review the logic above to help your child master the concept of pre calculus trig identities worksheet.
