If `tantheta=cottheta" and "0^(@) le theta le 90^(@)`, state the value of `theta`.

Answer

To solve the equation \( \tan \theta = \cot \theta \) for \( 0^\circ \leq \theta \leq 90^\circ \), we can follow these steps:

### Step 1: Understand the relationship between tangent and cotangent
We know that:
\[ \tan \theta = \frac{\sin \theta}{\cos \theta} \]
and
\[ \cot \theta = \frac{\cos \theta}{\sin \theta} \]
Given that \( \tan \theta = \cot \theta \), we can set these two expressions equal to each other:
\[ \frac{\sin \theta}{\cos \theta} = \frac{\cos \theta}{\sin \theta} \]

### Step 2: Cross-multiply
Cross-multiplying gives us:
\[ \sin^2 \theta = \cos^2 \theta \]

### Step 3: Rearrange the equation
We can rearrange the equation:
\[ \sin^2 \theta - \cos^2 \theta = 0 \]

### Step 4: Use the Pythagorean identity
Using the identity \( \sin^2 \theta + \cos^2 \theta = 1 \), we can express one function in terms of the other. However, in this case, we can also directly use the fact that:
\[ \sin^2 \theta = \cos^2 \theta \]
This implies:
\[ \frac{\sin^2 \theta}{\cos^2 \theta} = 1 \]
Thus,
\[ \tan^2 \theta = 1 \]

### Step 5: Solve for \( \tan \theta \)
Taking the square root of both sides gives us:
\[ \tan \theta = \pm 1 \]

### Step 6: Determine the angle \( \theta \)
Since we are considering the interval \( 0^\circ \leq \theta \leq 90^\circ \), we only consider the positive value:
\[ \tan \theta = 1 \]
The angle for which \( \tan \theta = 1 \) is:
\[ \theta = 45^\circ \]

### Final Answer
Thus, the value of \( \theta \) is:
\[ \theta = 45^\circ \]

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