If `tan theta = 1/(sqrt(11)) and 0 < theta < pi/2`, then the value of `("cosec^2 theta - sec^2 theta)/(cosec^2 theta + sec^2 theta)` is

Options

A. `3/4`
B. `4/5`
C. `5/6`
D. `6/7`

Answer

To solve the problem, we need to find the value of \(\frac{\csc^2 \theta - \sec^2 \theta}{\csc^2 \theta + \sec^2 \theta}\) given that \(\tan \theta = \frac{1}{\sqrt{11}}\) and \(0 < \theta < \frac{\pi}{2}\).

### Step 1: Find \(\sin \theta\) and \(\cos \theta\)

Given \(\tan \theta = \frac{1}{\sqrt{11}}\), we can express this as:

\[ \tan \theta = \frac{\text{opposite}}{\text{adjacent}} = \frac{1}{\sqrt{11}} \]

This implies that in a right triangle, the opposite side is \(1\) and the adjacent side is \(\sqrt{11}\).

Using the Pythagorean theorem, we find the hypotenuse \(h\):

\[ h = \sqrt{1^2 + (\sqrt{11})^2} = \sqrt{1 + 11} = \sqrt{12} = 2\sqrt{3} \]

Now we can find \(\sin \theta\) and \(\cos \theta\):

\[ \sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{1}{2\sqrt{3}} = \frac{\sqrt{3}}{6} \]

\[ \cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{\sqrt{11}}{2\sqrt{3}} = \frac{\sqrt{33}}{6} \]

### Step 2: Calculate \(\csc^2 \theta\) and \(\sec^2 \theta\)

Now we can find \(\csc^2 \theta\) and \(\sec^2 \theta\):

\[ \csc^2 \theta = \frac{1}{\sin^2 \theta} = \frac{1}{\left(\frac{1}{2\sqrt{3}}\right)^2} = \frac{1}{\frac{1}{12}} = 12 \]

\[ \sec^2 \theta = \frac{1}{\cos^2 \theta} = \frac{1}{\left(\frac{\sqrt{11}}{2\sqrt{3}}\right)^2} = \frac{1}{\frac{11}{12}} = \frac{12}{11} \]

### Step 3: Substitute into the expression

Now we substitute \(\csc^2 \theta\) and \(\sec^2 \theta\) into the expression:

\[ \frac{\csc^2 \theta - \sec^2 \theta}{\csc^2 \theta + \sec^2 \theta} = \frac{12 - \frac{12}{11}}{12 + \frac{12}{11}} \]

### Step 4: Simplify the numerator and denominator

First, we simplify the numerator:

\[ 12 - \frac{12}{11} = \frac{12 \cdot 11}{11} - \frac{12}{11} = \frac{132 - 12}{11} = \frac{120}{11} \]

Now we simplify the denominator:

\[ 12 + \frac{12}{11} = \frac{12 \cdot 11}{11} + \frac{12}{11} = \frac{132 + 12}{11} = \frac{144}{11} \]

### Step 5: Final calculation

Now we have:

\[ \frac{\frac{120}{11}}{\frac{144}{11}} = \frac{120}{144} = \frac{5}{6} \]

Thus, the final answer is:

\[ \frac{5}{6} \]