Simplify (sin A) / (1 + cos A) + cot A, where A is an acute angle.

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Given expression: (sin A) / (1 + cos A) + cot A

Step 1: Recall that cot A = cos A / sin A.

Rewrite the expression:
= (sin A) / (1 + cos A) + (cos A) / (sin A)

Step 2: Find a common denominator for the two terms:
Common denominator = (1 + cos A)(sin A)

Rewrite each term with the common denominator:
= [sin A * sin A] / [(1 + cos A)(sin A)] + [cos A * (1 + cos A)] / [(1 + cos A)(sin A)]
= (sin^2 A + cos A + cos^2 A) / [(1 + cos A)(sin A)]

Step 3: Use the Pythagorean identity sin^2 A + cos^2 A = 1:
= (1 + cos A) / [(1 + cos A)(sin A)]

Step 4: Cancel (1 + cos A) from numerator and denominator:
= 1 / sin A

Step 5: Recall that 1 / sin A = cosec A.

But the expression simplifies to cosec A, which is option 4.

The final simplified form is cosec A.

Therefore, the correct answer is option 4: cosec A.

Answer

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