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प्रश्न
If 2 tan–1(cos θ) = tan–1(2 cosec θ), then show that θ = π 4, where n is any integer.
उत्तर
2 tan–1(cos θ) = tan–1(2 cosec θ)
⇒ `tan^-1 ((2costheta)/(1 - cos^2 theta)) = tan^-1(2 "cosec" theta)` ......`[because 2tan^-1x = tan^-1 (2x)/(1 - x^2)]`
⇒ `(2costheta)/(1 - cos^2theta)` = 2 cosec θ
⇒ `(2costheta)/(sin^2theta) = 2/sintheta`
⇒ cos θ sin θ = sin2θ
⇒ cos θ sin θ – sin2θ = 0
⇒ sin θ(cos θ – sin θ) = 0
⇒ sin θ = 0 or cos θ – sin θ = 0
⇒ sin θ = 0 or 1 – tan θ = 0
⇒ θ = 0 or tan θ = 1
⇒ θ = 0° or θ = `pi/4`
Hence proved.
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