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Sum of Cosines of Angles in Arithmetic Progression

The value of ...

Question

The value of $(1 + c o t θ - c o s e c θ) (1 + t a n θ + s e c θ)$ is

A

1

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B

2

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C

4

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Solution

The correct option is B 2
$(1 + c o t θ - c o s e c θ) (1 + t a n θ + s e c θ)$
$= 1 + t a n θ + s e c θ + c o t θ + c o t θ t a n θ + c o t θ s e c θ - c o s e c θ - c o s e c θ t a n θ - c o s e c θ s e c θ$
$= 1 + t a n θ + s e c θ + c o t θ + (\frac{1}{t a n θ} \times t a n θ) + (\frac{c o s θ}{s i n θ} \times \frac{1}{c o s θ}) - c o s e c θ - (\frac{1}{s i n θ} \times \frac{s i n θ}{c o s θ}) - c o s e c θ s e c θ$
$= 1 + t a n θ + s e c θ + c o t θ + 1 + \frac{1}{s i n θ} - c o s e c θ - \frac{1}{c o s θ} - c o s e c θ s e c θ$
$= 1 + t a n θ + s e c θ + c o t θ + 1 + c o s e c θ - c o s e c θ - s e c θ - c o s e c θ s e c θ$
$= 2 + t a n θ + c o t θ - s e c θ c o s e c θ$
$= 2 + \frac{s i n θ}{c o s θ} + \frac{c o s θ}{s i n θ} - \frac{1}{s i n θ c o s θ}$
$= \frac{2 s i n θ c o s θ + s i n^{2} θ + c o s^{2} θ - 1}{s i n θ c o s θ} = 2 \frac{s i n θ c o s θ}{s i n θ c o s θ} = 2$

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