If `tantheta=sinthetaand0^(@)lethetale90^(@)`,then find the value of `theta`.

Answer

To solve the equation \( \tan \theta = \sin \theta \) where \( 0^\circ < \theta < 90^\circ \), we can follow these steps:

### Step 1: Write the equation in terms of sine and cosine
We know that:
\[ \tan \theta = \frac{\sin \theta}{\cos \theta} \]
So, we can rewrite the equation:
\[ \frac{\sin \theta}{\cos \theta} = \sin \theta \]

### Step 2: Cross-multiply to eliminate the fraction
To eliminate the fraction, we can cross-multiply:
\[ \sin \theta = \sin \theta \cdot \cos \theta \]

### Step 3: Rearrange the equation
Rearranging gives us:
\[ \sin \theta - \sin \theta \cdot \cos \theta = 0 \]
Factoring out \( \sin \theta \):
\[ \sin \theta (1 - \cos \theta) = 0 \]

### Step 4: Set each factor to zero
This gives us two cases to consider:
1. \( \sin \theta = 0 \)
2. \( 1 - \cos \theta = 0 \)

### Step 5: Solve the first case
For \( \sin \theta = 0 \):
- The solutions are \( \theta = 0^\circ, 180^\circ, \ldots \)
- However, since \( 0^\circ < \theta < 90^\circ \), this case does not provide a valid solution.

### Step 6: Solve the second case
For \( 1 - \cos \theta = 0 \):
\[ \cos \theta = 1 \]
- The solution for this is \( \theta = 0^\circ \).

### Step 7: Check the validity of the solution
Since \( 0^\circ < \theta < 90^\circ \), the only solution that fits is:
\[ \theta = 0^\circ \]

### Final Answer
Thus, the value of \( \theta \) is:
\[ \theta = 0^\circ \]