1-sin2θ1+cosθ+1-1+cosθsinθ-sinθ1-cosθ-1secθ=

1-sin2θ1+cosθ+1-1+cosθsinθ-sinθ1-cosθ-1secθ =1-1-cos2θ1+cosθ+1+cosθsinθ-sinθ1-cosθ×1+cosθ1+cosθ-cosθ =1-(1-cosθ)+1+cosθsinθ-1+cosθsinθ-cosθ =1-1+cosθ-cosθ=0

1-sin2θ1+cosθ+1-1+cosθsinθ-sinθ1-cosθ-1secθ=

1-sin2θ1+cosθ+1-1+cosθsinθ-sinθ1-cosθ-1secθ =1-1-cos2θ1+cosθ+1+cosθsinθ-sinθ1-cosθ×1+cosθ1+cosθ-cosθ =1-(1-cosθ)+1+cosθsinθ-1+cosθsinθ-cosθ =1-1+cosθ-cosθ=0

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$1 - \frac{\sin^{2} θ}{1 + cosθ} + 1 - \frac{1 + cosθ}{sinθ} - \frac{sinθ}{1 - cosθ} - \frac{1}{secθ} =$

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$Sinθ$

$Cosθ$

answer is C.

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Detailed Solution

$1 - \frac{\sin^{2} θ}{1 + cosθ} + 1 - \frac{1 + cosθ}{sinθ} - \frac{sinθ}{1 - cosθ} - \frac{1}{secθ} = 1 - \frac{1 - \cos^{2} θ}{1 + cosθ} + \frac{1 + cosθ}{sinθ} - \frac{sinθ}{1 - cosθ} \times \frac{1 + cosθ}{1 + cosθ} - cosθ = 1 - (1 - cosθ) + \frac{1 + cosθ}{sinθ} - \frac{1 + cosθ}{sinθ} - cosθ = 1 - 1 + cosθ - cosθ = 0$