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Which of the following is the value of the expression $cot A + cosec A − 1 cot A − cosec A + 1$ ?

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1 + cos A sin A

1 − cos A sin A

− 1 + cos A sin A

− 1 − cos A sin A

answer is A.

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Detailed Solution

The given expression is,

cot A + cosec A − 1 cot A − cosec A + 1

We know the trigonometric identities,

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

Therefore we have,

⇒ cot A + c o s e c A − 1 cot A − c o s e c A + 1

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

⇒ cos A + 1 − sin A cos A − 1 + sin A

⇒ cos A + 1 − sin A cos A + sin A − 1 × cos A + sin A + 1 cos A + sin A + 1

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

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

⇒ cos 2 A + cos 2 A + 2 cos A 2 sin A cos A

⇒ 2 cos A cos A + 1 2 sin A cos A

⇒ 1 + cos A sin A

Therefore, the value of the trigonometric expression

cot A + cosec A − 1 cot A − cosec A + 1 = 1 + cos A sin A .

Hence, option (1) is correct.

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