If `3+cos^(2)theta=3(cot^2theta+sin^2theta).0^(@)ltthetalt90^(@)`, then what is the value of `(costheta+2sintheta)`?

Options

A. `3sqrt2`
B. `(sqrt3+2)/2`
C. `(2sqrt3+1)/2`
D. `(3sqrt3+1)/2`

Answer

To solve the equation \(3 + \cos^2 \theta = 3(\cot^2 \theta + \sin^2 \theta)\) for \(0^\circ < \theta < 90^\circ\) and find the value of \(\cos \theta + 2 \sin \theta\), we can follow these steps:

### Step 1: Rewrite the equation
We start with the original equation:
\[ 3 + \cos^2 \theta = 3(\cot^2 \theta + \sin^2 \theta) \]
Recall that \(\cot^2 \theta = \frac{\cos^2 \theta}{\sin^2 \theta}\). Substituting this into the equation gives:
\[ 3 + \cos^2 \theta = 3\left(\frac{\cos^2 \theta}{\sin^2 \theta} + \sin^2 \theta\right) \]

### Step 2: Simplify the right-hand side
The right-hand side can be simplified:
\[ 3 + \cos^2 \theta = 3\left(\frac{\cos^2 \theta + \sin^4 \theta}{\sin^2 \theta}\right) \]
This can be rewritten as:
\[ 3 + \cos^2 \theta = \frac{3\cos^2 \theta + 3\sin^4 \theta}{\sin^2 \theta} \]

### Step 3: Multiply through by \(\sin^2 \theta\)
To eliminate the fraction, multiply both sides by \(\sin^2 \theta\):
\[ (3 + \cos^2 \theta)\sin^2 \theta = 3\cos^2 \theta + 3\sin^4 \theta \]
Expanding the left-hand side gives:
\[ 3\sin^2 \theta + \cos^2 \theta \sin^2 \theta = 3\cos^2 \theta + 3\sin^4 \theta \]

### Step 4: Rearranging the equation
Rearranging the equation leads to:
\[ 3\sin^2 \theta - 3\sin^4 \theta + \cos^2 \theta \sin^2 \theta - 3\cos^2 \theta = 0 \]
Factoring out common terms:
\[ 3\sin^2 \theta(1 - \sin^2 \theta) + \cos^2 \theta(\sin^2 \theta - 3) = 0 \]
Using the identity \(1 - \sin^2 \theta = \cos^2 \theta\):
\[ 3\sin^2 \theta \cos^2 \theta + \cos^2 \theta(\sin^2 \theta - 3) = 0 \]

### Step 5: Factor out \(\cos^2 \theta\)
Factoring out \(\cos^2 \theta\):
\[ \cos^2 \theta(3\sin^2 \theta + \sin^2 \theta - 3) = 0 \]
This gives us two cases:
1. \(\cos^2 \theta = 0\) (not possible since \(\theta\) is between \(0^\circ\) and \(90^\circ\))
2. \(4\sin^2 \theta - 3 = 0\)

### Step 6: Solve for \(\sin^2 \theta\)
Solving \(4\sin^2 \theta - 3 = 0\) gives:
\[ \sin^2 \theta = \frac{3}{4} \]
Thus, \(\sin \theta = \frac{\sqrt{3}}{2}\).

### Step 7: Find \(\cos \theta\)
Using the Pythagorean identity:
\[ \cos^2 \theta = 1 - \sin^2 \theta = 1 - \frac{3}{4} = \frac{1}{4} \]
Thus, \(\cos \theta = \frac{1}{2}\).

### Step 8: Calculate \(\cos \theta + 2 \sin \theta\)
Now we calculate:
\[ \cos \theta + 2 \sin \theta = \frac{1}{2} + 2 \cdot \frac{\sqrt{3}}{2} = \frac{1}{2} + \sqrt{3} \]

### Final Answer
The value of \(\cos \theta + 2 \sin \theta\) is:
\[ \frac{1}{2} + \sqrt{3} \]