Simplify `sqrt((1 - sin^(2) theta) div (1- cos^(2) theta))`

Options

A. `Cot theta`
B. `Tan theta`
C. `Sec theta`
D. `"Cosec " theta`

Answer

To simplify the expression \(\sqrt{\frac{1 - \sin^2 \theta}{1 - \cos^2 \theta}}\), we can follow these steps:

### Step 1: Identify the Trigonometric Identities
We know from trigonometric identities that:
\[ \sin^2 \theta + \cos^2 \theta = 1 \]
From this identity, we can express \(1 - \sin^2 \theta\) and \(1 - \cos^2 \theta\).

### Step 2: Rewrite the Numerator
Using the identity, we can rewrite the numerator:
\[ 1 - \sin^2 \theta = \cos^2 \theta \]

### Step 3: Rewrite the Denominator
Similarly, we can rewrite the denominator:
\[ 1 - \cos^2 \theta = \sin^2 \theta \]

### Step 4: Substitute Back into the Expression
Now, substituting these identities back into the original expression gives us:
\[ \sqrt{\frac{\cos^2 \theta}{\sin^2 \theta}} \]

### Step 5: Simplify the Fraction
The fraction \(\frac{\cos^2 \theta}{\sin^2 \theta}\) can be simplified to:
\[ \left(\frac{\cos \theta}{\sin \theta}\right)^2 \]

### Step 6: Take the Square Root
Taking the square root of the simplified fraction results in:
\[ \frac{\cos \theta}{\sin \theta} = \cot \theta \]

### Final Result
Thus, the simplified form of the original expression is:
\[ \sqrt{\frac{1 - \sin^2 \theta}{1 - \cos^2 \theta}} = \cot \theta \]

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