Find the value of `(sintheta)/(sqrt(1-sin^(2)theta))`

Answer

To find the value of \(\frac{\sin \theta}{\sqrt{1 - \sin^2 \theta}}\), we can follow these steps:

### Step 1: Recognize the Pythagorean Identity
We know from trigonometric identities that:
\[ \sin^2 \theta + \cos^2 \theta = 1 \]
This implies that:
\[ 1 - \sin^2 \theta = \cos^2 \theta \]

### Step 2: Substitute into the Expression
Now, we can substitute \(1 - \sin^2 \theta\) in our original expression:
\[ \frac{\sin \theta}{\sqrt{1 - \sin^2 \theta}} = \frac{\sin \theta}{\sqrt{\cos^2 \theta}} \]

### Step 3: Simplify the Square Root
The square root of \(\cos^2 \theta\) is \(|\cos \theta|\). However, since we are generally considering angles where \(\cos \theta\) is positive (like in the first quadrant), we can simplify:
\[ \sqrt{\cos^2 \theta} = \cos \theta \]

### Step 4: Final Simplification
Now, we can simplify the expression:
\[ \frac{\sin \theta}{\cos \theta} = \tan \theta \]

### Conclusion
Thus, the value of \(\frac{\sin \theta}{\sqrt{1 - \sin^2 \theta}}\) is:
\[ \tan \theta \]

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