For the every `theta,0ltthetale90`, find the values of the following angle
`tantheta+cottheta=2`

Options

A. `30^@`
B. `45^@`
C. `60^@`
D. `55^@`

Answer

To solve the equation \( \tan \theta + \cot \theta = 2 \) for \( 0 < \theta < 90 \), we can follow these steps:

### Step 1: Rewrite the equation
We know that:
\[ \tan \theta = \frac{\sin \theta}{\cos \theta} \quad \text{and} \quad \cot \theta = \frac{\cos \theta}{\sin \theta} \]
Thus, we can rewrite the equation as:
\[ \frac{\sin \theta}{\cos \theta} + \frac{\cos \theta}{\sin \theta} = 2 \]

### Step 2: Find a common denominator
The common denominator for the left-hand side is \( \sin \theta \cos \theta \). Therefore, we can rewrite the equation as:
\[ \frac{\sin^2 \theta + \cos^2 \theta}{\sin \theta \cos \theta} = 2 \]

### Step 3: Use the Pythagorean identity
Using the Pythagorean identity \( \sin^2 \theta + \cos^2 \theta = 1 \), we can substitute this into our equation:
\[ \frac{1}{\sin \theta \cos \theta} = 2 \]

### Step 4: Cross-multiply
Cross-multiplying gives us:
\[ 1 = 2 \sin \theta \cos \theta \]

### Step 5: Use the double angle identity
We know that \( \sin(2\theta) = 2 \sin \theta \cos \theta \). Thus, we can rewrite the equation as:
\[ 1 = \sin(2\theta) \]

### Step 6: Solve for \( 2\theta \)
The equation \( \sin(2\theta) = 1 \) implies:
\[ 2\theta = 90^\circ \]
Thus,
\[ \theta = 45^\circ \]

### Conclusion
The value of \( \theta \) that satisfies the equation \( \tan \theta + \cot \theta = 2 \) is:
\[ \theta = 45^\circ \]