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If $cosec A − cot A = q,$ then which of the following is the value of the expression $q 2 − 1 q 2 + 1 + cos A$ ?

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answer is A.

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Detailed Solution

The given equation is

cosec A − cot A = q

Apply the following:

cot θ = cos θ sin θ, c o s e c θ = 1 sin θ, sin 2 θ = 1 − cos 2 θ

Now, we have,

⇒ q 2 − 1 q 2 + 1 + cos A

⇒ c o s e c A − cot A 2 − 1 c o s e c A − cot A 2 + 1 + cos A ∵ cosec A − cot A = q

⇒ 1 sin A − cos A sin A 2 − 1 1 sin A − cos A sin A 2 + 1 + cos A

⇒ 1 − cos A 2 − sin 2 A 1 − cos A 2 + sin 2 A + cos A

⇒ 1 + cos 2 A − 2 cos A − sin 2 A 1 + cos 2 A − 2 cos A + sin 2 A + cos A

⇒ 1 − sin 2 A + cos 2 A − 2 cos A 1 + cos 2 A + sin 2 A − 2 cos A + cos A

⇒ cos 2 A + cos 2 A − 2 cos A 1 + 1 − 2 cos A + cos A

⇒ − 2 cos A 1 − cos A 2 1 − cos A + cos A

⇒ − cos A + cos A = 0

Therefore, the value of the expression

q 2 − 1 q 2 + 1 + cos A = 0.

Hence, option (1) is correct.

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