Problem 1 : Prove : (1 - cos2θ) ⋅ csc2θ = 1 Problem 2 : Prove : secθ ⋅ √(1 - sin2θ) = 1 Problem 3 : Prove : tanθ ⋅ sinθ + cosθ = secθ Problem 4 : Prove : (1 - cosθ) ⋅ (1 + cosθ) ⋅ (1 + cot2θ) = 1 Problem 5 : Prove : cotθ + tanθ = secθ ⋅ cscθ Problem 6 : Prove : cosθ/(1 - tanθ) + sinθ/(1 - cotθ) = sinθ + cosθ Problem 7 : Prove : tan4θ + tan2θ = sec4θ - sec2θ Problem 8 : Prove : √{(secθ – 1)/(secθ + 1)} = cscθ - cotθ Problem 9 : Prove : (1 - sinA)/(1 + sinA) = (secA - tanA)2 Problem 10 : Prove : (tanθ + secθ - 1)/(tanθ - secθ + 1) = (1 + sinθ)/cosθ Answers1. Answer : (1 - cos2θ) ⋅ csc2θ = 1 Let A = (1 - cos2θ) ⋅ csc2θ
and B = 1. Because, sin2θ + cos2θ = 1, we have sin2θ = 1 - cos2θ Then, we have A = sin2θ ⋅ csc2θ We know that csc2θ = 1/sin2θ A = sin2θ ⋅ 1/sin2θ A = 1 A = B Proved 2. Answer : secθ ⋅ √(1 - sin2θ) = 1 Let A = secθ ⋅ √(1
- sin2θ) and B = 1. Because sin2θ + cos2θ = 1, we have cos2θ = 1 - sin2θ. A = secθ ⋅ √cos2θ A = secθ ⋅ cosθ A = secθ ⋅ 1/secθ A = 1 A = B Proved 3. Answer : tanθ ⋅ sinθ + cosθ = secθ Let A = tanθ ⋅ sin θ + cos θ and B = sec θ. A = tanθ ⋅ sin θ + cos θ A = (sinθ/cosθ) ⋅ sinθ + cosθ A = (sin2θ/cosθ) + cosθ A = (sin2θ + cos2θ)/cosθ A = 1/cosθ A = secθ A = B Proved 4. Answer : (1 - cosθ) ⋅ (1 + cosθ) ⋅ (1 + cot2θ) = 1 Let A = (1 - cosθ) ⋅ (1 + cosθ) ⋅ (1 + cot2θ) and B = 1. A = (1 - cosθ) ⋅ (1 + cosθ) ⋅ (1 + cot2θ) A = (1 - cos2θ) ⋅ (1 + cot2θ) Because, sin2θ + cos2θ = 1, we have sin2θ = 1 - cos2θ. A = sin2θ ⋅ (1 + cot2θ) A = sin2θ + sin2θ ⋅ cos2θ A = sin2θ + sin2θ ⋅ (cos2θ/sin2θ) A = sin2θ + cos2θ A = 1 A = B Proved 5. Answer : cotθ + tanθ = secθ ⋅ cscθ Let A = cotθ + tanθ and B = secθ ⋅ cscθ. A = cotθ + tanθ A = cosθ/sinθ + sinθ/cosθ A = (cos2θ + sin2θ)/(sinθ ⋅ cosθ) (Because, cos2θ + sin2θ = 1) A = 1/(sinθ ⋅ cosθ) A = (1/cos θ) ⋅ (1/sin θ) A = secθ ⋅ cscθ A = B Proved 6. Answer : cosθ/(1 - tanθ) + sinθ/(1 - cotθ) = sinθ + cosθ Let A = cosθ/(1 - tanθ) + sinθ/(1 - cotθ) and B = sinθ + cosθ. A = cosθ/{1 - (sinθ/cos θ)} + sinθ/{1 - (cosθ/sinθ)} A = cosθ/{(cosθ - sinθ)/cosθ} + sinθ/{(sinθ - cosθ/sinθ)} A = cos2θ/(cosθ - sinθ) + sin2θ/(sinθ - cosθ) A = cos2θ/(cosθ - sinθ) - sin2θ/(cosθ - sinθ) A = (cos2θ - sin2θ)/(cosθ - sinθ) A = [(cosθ + sinθ)(cosθ - sinθ)]/(cosθ - sinθ) A = cosθ + sinθ A = B Proved 7. Answer : tan4θ + tan2θ = sec4θ - sec2θ Let A = tan2θ + tan2θ and B = sec2θ - sec2θ. A = tan2θ (tan2θ + 1) A = (sec2θ - 1)(tan2θ + 1) [Because tan2θ = sec2θ – 1] A = (sec2θ - 1) ⋅ sec2θ [Because, tan2θ + 1 = sec2θ] A = sec4θ - sec2θ A = B Proved 8. Answer : √{(secθ – 1)/(secθ + 1)} = cscθ - cotθ Let A = √{(secθ – 1)/(secθ + 1)} and B = cosecθ - cotθ. A = √{(secθ – 1)/(secθ + 1)} A = √[{(secθ - 1) (secθ - 1)}/{(secθ + 1) (secθ - 1)}] [Multiplying numerator and denominator by (secθ - l) inside radical sign] A = √{(secθ - 1)2/(sec2θ - 1)} A = √{(secθ -1)2/tan2θ} [Because, sec2 θ = 1 + tan2 θ ⇒ sec2θ - 1 = tan2θ] A = (secθ – 1)/tanθ A = (secθ/tanθ) – (1/tanθ) A = {(1/cosθ)/(sinθ/cos θ)} - cotθ A = {(1/cosθ) ⋅ (cosθ/sinθ)} - cotθ A = (1/sinθ) -
cotθ A = cosecθ - cotθ A = B Proved 9. Answer : (1 - sinA)/(1 + sinA) = (secA - tanA)2 Let A = (1 - sinA)/(1 + sinA) and B = (secA - tanA)2. A = (1 - sinA)/(1 + sinA) A = (1 - sinA)2/(1 - sinA)(1 + sinA) [Multiply both numerator and denominator by (1 - sin A) A = (1 - sinA)2/(1 - sin2A) A = (1 - sinA)2/(cos2A) [Because, sin2θ + cos2θ = 1 ⇒ cos2θ = 1 - sin2θ] A = {(1 - sinA)/cosA}2 A = (1/cosA - sinA/cosA)2 A = (secA – tanA)2 A = B Proved 10. Answer : (tanθ + secθ - 1)/(tanθ - secθ + 1) = (1 + sinθ)/cosθ Let
A = (tanθ + secθ - 1)/(tanθ - secθ + 1) and B = (1 + sinθ)/cosθ. A = (tanθ + secθ - 1)/(tanθ - secθ + 1) A = [(tanθ + secθ) - (sec2θ - tan2θ)]/(tanθ - secθ + 1) [Because sec2θ - tan2θ = 1] A = {(tanθ + secθ) -
(secθ + tanθ)(secθ - tanθ)}/(tanθ - secθ + 1) A = {(tanθ + secθ) (1 - secθ + tanθ)}/(tanθ - secθ + 1) A = {(tanθ + secθ) (tanθ - secθ + 1)}/(tanθ - secθ + 1) A = tanθ + secθ A = (sinθ/cosθ) + (1/cosθ) A = (sinθ + 1)/cosθ A = (1 + sinθ)/cosθ A = B Proved Kindly mail your feedback to We always appreciate your feedback. ©All rights reserved. onlinemath4all.com |