Express in simplest form: `(sintheta)/(sqrt(1-sin^2theta))=`

Options

A. `cottheta`
B. `sqrtsintheta`
C. `sintheta//sqrtcostheta`
D. `tantheta`

Answer

To express \(\frac{\sin \theta}{\sqrt{1 - \sin^2 \theta}}\) in its simplest form, follow these steps:

### Step-by-Step Solution:

1. **Identify the expression**:
We start with the expression \(\frac{\sin \theta}{\sqrt{1 - \sin^2 \theta}}\).

2. **Use the Pythagorean identity**:
Recall the trigonometric identity:
\[ \sin^2 \theta + \cos^2 \theta = 1 \]
From this, we can express \(\cos^2 \theta\) as:
\[ \cos^2 \theta = 1 - \sin^2 \theta \]

3. **Substitute in the expression**:
Substitute \(\cos^2 \theta\) into the expression:
\[ \sqrt{1 - \sin^2 \theta} = \sqrt{\cos^2 \theta} \]
Since \(\sqrt{\cos^2 \theta} = |\cos \theta|\), we can write:
\[ \frac{\sin \theta}{\sqrt{1 - \sin^2 \theta}} = \frac{\sin \theta}{|\cos \theta|} \]

4. **Consider the sign of \(\cos \theta\)**:
If we assume \(\theta\) is in a range where \(\cos \theta\) is positive (for example, \(0 \leq \theta < \frac{\pi}{2}\)), we can simplify further:
\[ \frac{\sin \theta}{\cos \theta} = \tan \theta \]

5. **Final answer**:
Thus, the simplest form of the expression is:
\[ \tan \theta \]