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Which of the following is the value of the expression $c o s e c θ − sin θ sec θ − cos θ$ ?

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sin θ cos θ

2 sin θ cos θ

sin 2 θ cos θ

sin θ cos 2 θ

answer is A.

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Detailed Solution

The given expression is,

c o s e c θ − sin θ sec θ − cos θ

Apply the following:

c o s e c θ = 1 sin θ, sec θ = 1 cos θ, sin 2 θ + cos 2 θ = 1

Therefore we have,

⇒ c o s e c θ − sin θ sec θ − cos θ

⇒ 1 sin θ − sin θ 1 cos θ − cos θ

⇒ 1 − sin 2 θ sin θ 1 − cos 2 θ cos θ

⇒ cos 2 θ sin θ sin 2 θ cos θ

⇒ sin θ cos θ

Therefore, the value of the expression

c o s e c θ − sin θ sec θ − cos θ = sin θ cos θ .

Hence, option (1) is correct.

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