(1+tanθ+secθ)(1+cotθ−cosecθ)=

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Question

(1 + tan θ + sec θ) (1 + cot θ - c o s e c θ) =

Options:

A . 0

B . 1

C . 2

D . -1

Answer: Option C
:
C

(1 + tan θ + sec θ) (1 + cot θ - c o s e c θ)

= (1 + \frac{sin θ}{cos θ} + \frac{1}{cos θ}) (1 + \frac{cos θ}{sin θ} - \frac{1}{sin θ})

= \frac{(cos θ + sin θ + 1)}{cos θ} \times \frac{(sin θ + cos θ - 1)}{sin θ}

= \frac{(cos θ + sin θ)^{2} - 1^{2}}{cos θ sin θ}

= \frac{({cos}^{2} θ + {sin}^{2} θ + 2 cos θ sin θ - 1)}{cos θ sin θ}

= \frac{(1 + 2 cos θ sin θ - 1)}{cos θ sin θ}

= \frac{2 cos θ sin θ}{cos θ sin θ} = 2

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