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If secθ + tangp , then find the value of cosecθ |
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Answer» Given: sec θ + tan θ = p ----- (i) We know that sec²θ - tan²θ = 1 ⇒ (secθ + tanθ)(secθ - tanθ) = 1 ⇒ (p)(secθ - tanθ) = 1 ⇒ secθ - tanθ = (1/p) ----- (ii) On solving (i) & (ii), we get ⇒ secθ + tanθ + secθ - tanθ = p + 1/p ⇒ 2secθ = p² + 1/p ⇒ secθ = (p² + 1)/2p ⇒ cosθ = (1/secθ) = 2p/p² + 1 Sin²θ = 1 - cos²θ = 1 - (2p/p² + 1)² = 1 - (4p²)/p⁴ + 1 + 2p² = (p⁴ + 1 + 2p² - 4p²)/p⁴ + 1 + 2p = (p⁴ + 1 - 2p²)/p⁴ + 1 + 2p = (p² - 1)²/(p² + 1)² sin θ= p² - 1/p² + 1. Now, We know that cosecθ = (1/sinθ) ⇒ (p² + 1)/p² - 1. |
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