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Which of the following is the value of the expression $sin A 1 + cos A + 1 + cos A sin A sin A 1 − cos A − 1 − cos A sin A$ ?

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5cosec A cot A

3cosec A cot A

2cosec A cot A

4 cosec A cot A

answer is D.

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Detailed Solution

The given expression is,

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

We know the trigonometric identities,

sin 2 A + cos 2 A = 1, cos e c A = 1 sin A, cot A = cos A sin A

Therefore we have,

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

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

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

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

⇒ 2 sin A × 2 cos A sin A

⇒ 4 cosec A cot A

Therefore, the value of the trigonometric expression

sin A 1 + cos A + 1 + cos A sin A sin A 1 − cos A − 1 − cos A sin A = 4 cosec A cot A .

Hence, option (4) is correct.

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