What is the simplified trigonometric form of (1 + cos A)(cosec A cot A) when it is expressed in terms of sine and cosine?

Introduction / Context:
This question requires you to simplify a trigonometric expression that mixes several functions: cosine, cosecant, and cotangent. The goal is to rewrite everything in terms of sine and cosine and then simplify as far as possible to obtain a compact algebraic form.

Given Data / Assumptions:

- Expression: (1 + cos A)(cosec A cot A)
- A is an angle where all trigonometric functions are defined and sine is non zero
- We want to express the result using only sine and cosine functions

Concept / Approach:
We use the definitions cosec A = 1 / sin A and cot A = cos A / sin A. By substituting these into the given expression, we obtain a product of algebraic fractions in terms of sine and cosine. Then we multiply and simplify to a single rational expression. This is a standard method to simplify mixed trigonometric expressions.

Step-by-Step Solution:
Step 1: Start with (1 + cos A)(cosec A cot A).Step 2: Replace cosec A with 1 / sin A and cot A with cos A / sin A. The expression becomes (1 + cos A) * (1 / sin A) * (cos A / sin A).Step 3: Combine the fractions inside the product: (1 / sin A) * (cos A / sin A) = cos A / sin^2 A.Step 4: Now the expression is (1 + cos A) * [cos A / sin^2 A].Step 5: Multiply the terms in the numerator: (1 + cos A) * cos A = cos A + cos^2 A. So the complete simplified form is (cos A + cos^2 A) / sin^2 A, which we can write neatly as (1 + cos A)cos A / sin^2 A.

Verification / Alternative check:
Choose a simple angle, for example A = 30 degrees. Compute the original expression numerically: (1 + cos 30°)(cosec 30° cot 30°). cos 30° = √3/2, sin 30° = 1/2, cosec 30° = 2, and cot 30° = √3. The original expression is (1 + √3/2)(2 * √3). Now compute the simplified form: (1 + cos A)cos A / sin^2 A at 30 degrees gives (1 + √3/2)(√3/2) / (1/2)^2. Since (1/2)^2 = 1/4, the denominator becomes 1/4, and the whole expression is (1 + √3/2)(√3/2) * 4, which equals (1 + √3/2)(2√3). Both versions produce the same numerical value, confirming the simplification.

Why Other Options Are Wrong:
- (1 − cos A)cos A / sin^2 A changes the sign in the first factor and does not match the original expression when expanded.
- (1 + cos A) / sin A corresponds to only part of the expression and ignores one factor of cos A and an extra factor of sin A in the denominator.
- cos A / sin^2 A arises from the product cosec A cot A alone, without the factor (1 + cos A).
- sin A / cos^2 A is unrelated to the given structure and does not match any correct simplification of the original expression.

Common Pitfalls:
Common mistakes include forgetting that cosec A and cot A each contribute a factor of 1 / sin A, leading to sin^2 A in the denominator, or mixing up cot A with tan A. Another frequent error is to stop simplifying too early, instead of combining everything into a single fraction. Carefully rewriting all functions in terms of sine and cosine and then simplifying step by step helps avoid these issues.

Final Answer:
The simplified form of (1 + cos A)(cosec A cot A) is (1 + cos A)cos A / sin^2 A.