If cos θ≠0 and sec θ-1=(2-1) tan θ then θ=

secθ-1=2-1tanθ 1-cosθ=2-1sinθ sinθ tanπ8+cosθ=1 cosθ cosπ8+sinθ sinπ8=cosπ8 cosθ-π8=cosπ8 θ-π8=2nπ±π8 θ=2nπ, θ=2nπ+π4

If cos θ≠0 and sec θ-1=(2-1) tan θ then θ=

secθ-1=2-1tanθ 1-cosθ=2-1sinθ sinθ tanπ8+cosθ=1 cosθ cosπ8+sinθ sinπ8=cosπ8 cosθ-π8=cosπ8 θ-π8=2nπ±π8 θ=2nπ, θ=2nπ+π4

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If $\cos θ \neq 0$ and $\sec θ - 1 = (\sqrt{2} - 1) \tan θ$ then $θ =$

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$nπ + \frac{π}{8}$ (or) $2 nπ, n \in Z$

$2 nπ + \frac{π}{6}$ (or) $2 nπ, n \in Z$

$2 nπ + \frac{π}{8}, n \in Z$

$2 nπ - \frac{π}{4}$ (or) $2 nπ, n \in Z$

answer is A.

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Detailed Solution

$secθ - 1 = (\sqrt{2} - 1) tanθ 1 - cosθ = (\sqrt{2} - 1) sinθ sinθ \tan \frac{π}{8} + cosθ = 1 cosθ \cos \frac{π}{8} + sinθ \sin \frac{π}{8} = \cos \frac{π}{8} \cos (θ - \frac{π}{8}) = \cos \frac{π}{8} θ - \frac{π}{8} = 2 nπ \pm \frac{π}{8} θ = 2 nπ, θ = 2 nπ + \frac{π}{4}$