If `alpha` is acute angle and `2sin^(2)alpha+15cos^(2)alpha=7`, then `cotalpha` equal to

Options

A. `(4)/(3)`
B. `(sqrt(5))/(2sqrt(2))`
C. `(2)/(sqrt(5))`
D. `(3)/(4)`

Answer

To solve the equation \(2\sin^2\alpha + 15\cos^2\alpha = 7\) and find the value of \(\cot\alpha\), we can follow these steps:

### Step 1: Rewrite the equation
We start with the equation:
\[ 2\sin^2\alpha + 15\cos^2\alpha = 7 \]
We know that \(\sin^2\alpha + \cos^2\alpha = 1\). Therefore, we can express \(\sin^2\alpha\) in terms of \(\cos^2\alpha\):
\[ \sin^2\alpha = 1 - \cos^2\alpha \]
Substituting this into the equation gives:
\[ 2(1 - \cos^2\alpha) + 15\cos^2\alpha = 7 \]

### Step 2: Simplify the equation
Expanding the equation:
\[ 2 - 2\cos^2\alpha + 15\cos^2\alpha = 7 \]
Combine like terms:
\[ 2 + 13\cos^2\alpha = 7 \]

### Step 3: Isolate \(\cos^2\alpha\)
Now, we isolate \(\cos^2\alpha\):
\[ 13\cos^2\alpha = 7 - 2 \]
\[ 13\cos^2\alpha = 5 \]
\[ \cos^2\alpha = \frac{5}{13} \]

### Step 4: Find \(\cos\alpha\)
Taking the square root of both sides (since \(\alpha\) is acute, we take the positive root):
\[ \cos\alpha = \sqrt{\frac{5}{13}} = \frac{\sqrt{5}}{\sqrt{13}} \]

### Step 5: Find \(\sin\alpha\)
Using the identity \(\sin^2\alpha + \cos^2\alpha = 1\):
\[ \sin^2\alpha = 1 - \cos^2\alpha = 1 - \frac{5}{13} = \frac{8}{13} \]
Taking the square root:
\[ \sin\alpha = \sqrt{\frac{8}{13}} = \frac{\sqrt{8}}{\sqrt{13}} = \frac{2\sqrt{2}}{\sqrt{13}} \]

### Step 6: Calculate \(\cot\alpha\)
The cotangent is given by:
\[ \cot\alpha = \frac{\cos\alpha}{\sin\alpha} = \frac{\frac{\sqrt{5}}{\sqrt{13}}}{\frac{2\sqrt{2}}{\sqrt{13}}} \]
This simplifies to:
\[ \cot\alpha = \frac{\sqrt{5}}{2\sqrt{2}} = \frac{\sqrt{5}}{2\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} = \frac{\sqrt{10}}{4} \]

### Final Answer
Thus, the value of \(\cot\alpha\) is:
\[ \cot\alpha = \frac{\sqrt{10}}{4} \]