What is the simplified value of `((cosec A)/(cot A + tan A )) ^(2)` ?

Options

A. `2 cos ^(2) A `
B. ` 1 - sin^(2) A `
C. `sec^(2) A`
D. sec A tan A

Answer

To simplify the expression \(\left(\frac{\csc A}{\cot A + \tan A}\right)^2\), we can follow these steps:

### Step-by-Step Solution:

1. **Rewrite the Trigonometric Functions**:
We start by rewriting the cosecant, cotangent, and tangent in terms of sine and cosine:
\[ \csc A = \frac{1}{\sin A}, \quad \cot A = \frac{\cos A}{\sin A}, \quad \tan A = \frac{\sin A}{\cos A} \]
Thus, we can rewrite the expression:
\[ \frac{\csc A}{\cot A + \tan A} = \frac{\frac{1}{\sin A}}{\frac{\cos A}{\sin A} + \frac{\sin A}{\cos A}} \]

2. **Combine the Denominator**:
The denominator \(\cot A + \tan A\) can be combined by finding a common denominator:
\[ \cot A + \tan A = \frac{\cos A}{\sin A} + \frac{\sin A}{\cos A} = \frac{\cos^2 A + \sin^2 A}{\sin A \cos A} \]
Using the Pythagorean identity \(\cos^2 A + \sin^2 A = 1\), we have:
\[ \cot A + \tan A = \frac{1}{\sin A \cos A} \]

3. **Substituting Back**:
Now substituting back into our expression:
\[ \frac{\csc A}{\cot A + \tan A} = \frac{\frac{1}{\sin A}}{\frac{1}{\sin A \cos A}} = \frac{1}{\sin A} \cdot \frac{\sin A \cos A}{1} = \cos A \]

4. **Square the Result**:
Now we need to square the result:
\[ \left(\cos A\right)^2 = \cos^2 A \]

5. **Final Simplification**:
We can also express \(\cos^2 A\) using the Pythagorean identity:
\[ \cos^2 A = 1 - \sin^2 A \]

Thus, the simplified value of \(\left(\frac{\csc A}{\cot A + \tan A}\right)^2\) is \(\cos^2 A\) or \(1 - \sin^2 A\).

### Final Answer:
\[ \cos^2 A \quad \text{or} \quad 1 - \sin^2 A \]