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Question
If cotθ + tanθ = 2cosecθ, then find the general value of θ.
Solution
Given that: cotθ + tanθ = 2cosecθ
⇒ `costheta/sintheta + sintheta/costheta = 2/sintheta`
⇒ `(cos^2theta + sin^2theta)/(sintheta cos theta) = 2/sintheta`
⇒ `1/(sintheta costheta) = 2/sintheta`
⇒ 2sinθ cosθ = sinθ
⇒ 2sinθ cosθ – sinθ = 0
⇒ sinθ(2cosθ – 1) = 0
⇒ sinθ ≠ 0 or 2cosθ – 1 = 0 or cosθ = `1/2`
⇒ cosθ = `cos pi/3`
∴ θ = `2"n"pi +- pi/3`
Hence, the general value of θ is `2"n"pi +- pi/3`.
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