What is the simplified value of `((1)/(cosectheta+cottheta))^(2)`?

Options

A. `cosectheta+tantheta`
B. `sintheta+costheta`
C. `((1-costheta))/((1+costheta))`
D. `((1-sintheta))/((1+sintheta))`

Answer

To simplify the expression \(\left(\frac{1}{\csc \theta + \cot \theta}\right)^2\), we can follow these steps:

### Step 1: Rewrite \(\csc \theta\) and \(\cot \theta\)
We know that:
\[ \csc \theta = \frac{1}{\sin \theta} \quad \text{and} \quad \cot \theta = \frac{\cos \theta}{\sin \theta} \]
Thus, we can rewrite the expression:
\[ \csc \theta + \cot \theta = \frac{1}{\sin \theta} + \frac{\cos \theta}{\sin \theta} = \frac{1 + \cos \theta}{\sin \theta} \]

### Step 2: Substitute back into the expression
Now substituting this back into our original expression:
\[ \frac{1}{\csc \theta + \cot \theta} = \frac{1}{\frac{1 + \cos \theta}{\sin \theta}} = \frac{\sin \theta}{1 + \cos \theta} \]

### Step 3: Square the expression
Now we square the expression:
\[ \left(\frac{\sin \theta}{1 + \cos \theta}\right)^2 = \frac{\sin^2 \theta}{(1 + \cos \theta)^2} \]

### Step 4: Use the Pythagorean identity
Using the Pythagorean identity \(\sin^2 \theta = 1 - \cos^2 \theta\), we can substitute:
\[ \frac{\sin^2 \theta}{(1 + \cos \theta)^2} = \frac{1 - \cos^2 \theta}{(1 + \cos \theta)^2} \]

### Step 5: Factor the numerator
The numerator can be factored as:
\[ 1 - \cos^2 \theta = (1 - \cos \theta)(1 + \cos \theta) \]
Thus, we have:
\[ \frac{(1 - \cos \theta)(1 + \cos \theta)}{(1 + \cos \theta)^2} \]

### Step 6: Simplify the expression
Now we can cancel \(1 + \cos \theta\) from the numerator and the denominator:
\[ \frac{1 - \cos \theta}{1 + \cos \theta} \]

### Final Answer
Thus, the simplified value of \(\left(\frac{1}{\csc \theta + \cot \theta}\right)^2\) is:
\[ \frac{1 - \cos \theta}{1 + \cos \theta} \]

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