Question 30

If sec θ + tan θ = p, then find the value of cosec θ

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Question 30 If sec θ + tan θ = p, then find the value of cosec θ Given sec θ + tan θ = p …(1 We know that 1 + tan2 θ = sec2 θ 1 = sec2 θ – tan2 θ sec2 θ – tan2 θ = 1 Using a2 – b2 = (a + b) (a – b) (sec θ + tan θ) (sec θ – tan θ) = 1 Putting sec θ + tan θ = p p × (sec θ – tan θ) = 1 (sec θ – tan θ) = 1/𝑝 Now, our equations are sec θ + tan θ = p …(1) (sec θ – tan θ) = 1/𝑝 …(2) Adding (1) & (2) (sec θ + tan θ) + (sec θ – tan θ) = p + 1/𝑝 2 sec θ = p + 1/𝑝 2 sec θ = (𝑝^2 + 1)/𝑝 sec θ = (𝑝^2 + 1)/2𝑝 1/cos⁡𝜃 = (𝑝^2 + 1)/2𝑝 Cross multiplying cos θ = 2𝑝/(𝑝^2 + 1) Now, to find cosec θ We first find sin θ We know that cos2 θ + sin2 θ = 1 Putting value of cos θ from (3) (2𝑝/(𝑝^2 + 1))^2 + sin2 θ = 1 (4𝑝^2)/(𝑝^2 + 1)^2 + sin2 θ = 1 sin2 θ = 1 – (4𝑝^2)/(𝑝^2 + 1)^2 sin2 θ = ((𝑝^2 + 1)^2 − 4𝑝^2)/(𝑝^2 + 1)^2 sin2 θ = ((𝑝^2 )^2 + 1^2 + 2𝑝^2 − 4𝑝^2)/(𝑝^2 + 1)^2 sin2 θ = ((𝑝^2 )^2 + 1^2 − 2𝑝^2)/(𝑝^2 + 1)^2 Using (a – b)2 = a2 + b2 – 2ab where a = p2 , b = 1 sin2 θ = (𝑝^2 − 1)^2/(𝑝^2 + 1)^2 sin2 θ = ((𝑝^2 − 1)/(𝑝^2 + 1))^2 Cancelling squares sin θ = ± (𝑝^2 − 1)/(𝑝^2 + 1) For + sin θ = (𝑝^2 − 1)/(𝑝^2 + 1) Therefore, cosec θ = 1/sin⁡𝜃 = (𝒑^𝟐 + 𝟏)/(𝒑^𝟐 − 𝟏) For – sin θ = (−(𝑝^2 − 1))/(𝑝^2 + 1) Therefore, cosec θ = 1/sin⁡𝜃 = (𝒑^𝟐 + 𝟏)/(𝒑^𝟐 − 𝟏)

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Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 8 years. He provides courses for Maths and Science at Teachoo. You can check his NCERT Solutions from Class 6 to 12.