`"If "y=sqrt((2(tanalpha+cotalpha))/(1+tan^(2)alpha)+(1)/(sin^(2)alpha))"when "alpha in ((3pi)/(4),pi)"then find "(dy)/(dalpha)" at" alpha=(5pi)/(6)`

Answer

To solve the problem, we need to find the derivative of the function \( y \) with respect to \( \alpha \) and evaluate it at \( \alpha = \frac{5\pi}{6} \). Let's break this down step by step.

### Step 1: Write the expression for \( y \)
The given expression is:
\[ y = \sqrt{\frac{2(\tan \alpha + \cot \alpha)}{1 + \tan^2 \alpha} + \frac{1}{\sin^2 \alpha}} \]

### Step 2: Simplify \( y \)
We know that:
- \( \tan \alpha = \frac{\sin \alpha}{\cos \alpha} \)
- \( \cot \alpha = \frac{\cos \alpha}{\sin \alpha} \)

Substituting these into the expression for \( y \):
\[ y = \sqrt{\frac{2\left(\frac{\sin \alpha}{\cos \alpha} + \frac{\cos \alpha}{\sin \alpha}\right)}{1 + \tan^2 \alpha} + \frac{1}{\sin^2 \alpha}} \]

### Step 3: Rewrite \( 1 + \tan^2 \alpha \)
Using the identity \( 1 + \tan^2 \alpha = \sec^2 \alpha \):
\[ y = \sqrt{\frac{2\left(\frac{\sin \alpha}{\cos \alpha} + \frac{\cos \alpha}{\sin \alpha}\right)}{\sec^2 \alpha} + \frac{1}{\sin^2 \alpha}} \]
This simplifies to:
\[ y = \sqrt{2\left(\frac{\sin^2 \alpha + \cos^2 \alpha}{\sin \alpha \cos \alpha}\right) + \frac{1}{\sin^2 \alpha}} \]
Since \( \sin^2 \alpha + \cos^2 \alpha = 1 \):
\[ y = \sqrt{\frac{2}{\sin \alpha \cos \alpha} + \frac{1}{\sin^2 \alpha}} \]

### Step 4: Find a common denominator
The common denominator for the terms inside the square root is \( \sin^2 \alpha \cos \alpha \):
\[ y = \sqrt{\frac{2\sin \alpha + 1}{\sin^2 \alpha \cos \alpha}} \]

### Step 5: Further simplify \( y \)
We can express \( y \) as:
\[ y = \sqrt{2\cot \alpha + \frac{1}{\sin^2 \alpha}} \]
This can be rewritten as:
\[ y = \sqrt{1 + \cot^2 \alpha} = |1 + \cot \alpha| \]

### Step 6: Determine the sign of \( y \)
Since \( \alpha \) is in the interval \( \left(\frac{3\pi}{4}, \pi\right) \), \( \cot \alpha \) is negative. Thus:
\[ y = - (1 + \cot \alpha) = -1 - \cot \alpha \]

### Step 7: Differentiate \( y \)
Now we differentiate \( y \):
\[ \frac{dy}{d\alpha} = -\frac{d}{d\alpha}(\cot \alpha) = -(-\csc^2 \alpha) = \csc^2 \alpha \]

### Step 8: Evaluate \( \frac{dy}{d\alpha} \) at \( \alpha = \frac{5\pi}{6} \)
At \( \alpha = \frac{5\pi}{6} \):
\[ \csc^2\left(\frac{5\pi}{6}\right) = \left(\frac{1}{\sin\left(\frac{5\pi}{6}\right)}\right)^2 = \left(\frac{1}{\frac{1}{2}}\right)^2 = 4 \]

### Final Answer
Thus, the value of \( \frac{dy}{d\alpha} \) at \( \alpha = \frac{5\pi}{6} \) is:
\[ \frac{dy}{d\alpha} = 4 \]