Let `x, y, in R` satisfy the condition such that sin x sin y + 3 cos y +4 sin y cos `x=sqrt(26)`. The value of `tan^(2)x + cot^(2)y` is equal to

Options

A. `9xx17`
B. 205
C. `(1)/(16)+(9)/(17)`
D. none of these

Explanation

To solve the problem, we start with the equation given:

\[ \sin x \sin y + 3 \cos y + 4 \sin y \cos x = \sqrt{26} \]

### Step 1: Rearranging the Equation
We can rearrange the equation by factoring out \(\sin y\):

\[ \sin y (\sin x + 4 \cos x) + 3 \cos y = \sqrt{26} \]

### Step 2: Finding Maximum Value
To find the maximum value of the expression \(\sin y (\sin x + 4 \cos x) + 3 \cos y\), we can use the fact that the maximum value of \(a \sin x + b \cos x\) is given by \(\sqrt{a^2 + b^2}\).

Here, let:
- \(a = 4\)
- \(b = 3\)

Thus, the maximum value of \(4 \cos x + 3 \sin y\) is:

\[ \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5 \]

### Step 3: Setting Up the Condition
For the maximum value to be achieved, we need:

\[ \sin y (\sin x + 4 \cos x) + 3 \cos y = 5 \]

Given that this maximum value must equal \(\sqrt{26}\), we can set up the equation:

\[ \sqrt{26} \leq 5 \]

### Step 4: Finding Values of \(\tan^2 x\) and \(\cot^2 y\)
From the earlier steps, we can deduce the relationships:

1. \(\sin x + 4 \cos x = k\) (where \(k\) is a constant)
2. \(\sin y = \frac{3 \cos y + \sqrt{26}}{\sin x + 4 \cos x}\)

Using the maximum values, we can find:

\[ \tan^2 x = \left(\frac{\sin x}{\cos x}\right)^2 = \left(\frac{1}{4}\right)^2 = \frac{1}{16} \]

And for \(\cot^2 y\):

\[ \cot^2 y = \left(\frac{\cos y}{\sin y}\right)^2 = \left(\frac{3}{\sqrt{17}}\right)^2 = \frac{9}{17} \]

### Step 5: Combining Results
Now we can combine the results:

\[ \tan^2 x + \cot^2 y = \frac{1}{16} + \frac{9}{17} \]

To add these fractions, we need a common denominator. The least common multiple of \(16\) and \(17\) is \(272\):

\[ \tan^2 x + \cot^2 y = \frac{17}{272} + \frac{144}{272} = \frac{161}{272} \]

### Final Result
Thus, the value of \(\tan^2 x + \cot^2 y\) is:

\[ \frac{161}{272} \]