What is simplified value of `((1)/("cosec" theta + cot theta)) ^(2)` ?

Options

A. `"cosec" theta + tan theta`
B. `sin theta + cos theta`
C. `(1 - cos theta)//(1 + cos theta)`
D. `(1 - sin theta) //(1 - cos theta)//(1 + cos theta)`

Answer

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

### Step 1: Rewrite the trigonometric functions
Recall the definitions of cosecant and cotangent:
- \(\csc \theta = \frac{1}{\sin \theta}\)
- \(\cot \theta = \frac{\cos \theta}{\sin \theta}\)

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, substitute this back into the original expression:
\[ \left(\frac{1}{\csc \theta + \cot \theta}\right)^2 = \left(\frac{1}{\frac{1 + \cos \theta}{\sin \theta}}\right)^2 \]

### Step 3: Simplify the fraction
This simplifies to:
\[ \left(\frac{\sin \theta}{1 + \cos \theta}\right)^2 \]

### Step 4: Expand the square
Now, we can expand the square:
\[ \left(\frac{\sin \theta}{1 + \cos \theta}\right)^2 = \frac{\sin^2 \theta}{(1 + \cos \theta)^2} \]

### Step 5: Use the Pythagorean identity
Recall that \(\sin^2 \theta = 1 - \cos^2 \theta\). Thus, we can write:
\[ \frac{\sin^2 \theta}{(1 + \cos \theta)^2} = \frac{1 - \cos^2 \theta}{(1 + \cos \theta)^2} \]

### Step 6: Factor the numerator
The numerator can be factored as:
\[ 1 - \cos^2 \theta = (1 - \cos \theta)(1 + \cos \theta) \]

### Step 7: Substitute back into the expression
Substituting this back gives:
\[ \frac{(1 - \cos \theta)(1 + \cos \theta)}{(1 + \cos \theta)^2} \]

### Step 8: Cancel out common terms
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:
\[ \frac{1 - \cos \theta}{1 + \cos \theta} \]