What is the value of (1 + tan θ + sec θ) (1 + cot θ – cosec θ)?

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-1

answer is C.

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Detailed Solution

Given expression is (1 + tan θ + sec θ) (1 + cot θ – cosec θ).
Expanding the given expression, we get
(1 + tan θ + sec θ) (1 + cot θ – cosec θ)

= 1 + c o t θ − cos e c θ + tan θ 1 + c o t θ − cos e c θ + s e c θ 1 + c o t θ − cos e c θ = 1 + c o t θ − cos e c θ + tan θ + tan θ c o t θ − tan θ cos e c θ + s e c θ + c o t θ s e c θ − s e c θ cos e c θ

We know that,

cos e c θ = 1 sin θ, sec θ = 1 cos θ, cot θ = cos θ sin θ, tan θ = sin θ cos θ

= 2 + cos 2 θ + sin 2 θ cos θ sin θ − 1 cos θ sin θ

We know the identity

cos 2 θ + sin 2 θ = 1

, we get

= 2 + 1 cos θ sin θ − 1 cos θ sin θ = 2

Thus, the value of (1 + tan θ + sec θ) (1 + cot θ – cosec θ) is 2.
Therefore, the correct option is 3.

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