Find the value of `(tan^(2)37(1)/(2)^(@)+1)/(tan^(2)37(1)/(2)^(@)-1)` .

Explanation

To find the value of the expression

\[ \frac{\tan^2(37.5^\circ) + 1}{\tan^2(37.5^\circ) - 1} \]

we can follow these steps:

### Step 1: Rewrite the Expression
We start by rewriting the expression using the angle \( 37.5^\circ \):
\[ \tan^2(37.5^\circ) = \tan^2\left(\frac{75^\circ}{2}\right) \]
Thus, our expression becomes:
\[ \frac{\tan^2\left(\frac{75^\circ}{2}\right) + 1}{\tan^2\left(\frac{75^\circ}{2}\right) - 1} \]

### Step 2: Use the Identity
We know from trigonometric identities that:
\[ \tan^2\theta + 1 = \sec^2\theta \quad \text{and} \quad \tan^2\theta - 1 = \sec^2\theta - 2 \]
So we can rewrite the expression as:
\[ \frac{\sec^2\left(\frac{75^\circ}{2}\right)}{\sec^2\left(\frac{75^\circ}{2}\right) - 2} \]

### Step 3: Simplify the Expression
Now, we can simplify the expression:
\[ \frac{\sec^2\left(\frac{75^\circ}{2}\right)}{\sec^2\left(\frac{75^\circ}{2}\right) - 2} = \frac{1}{1 - \frac{2}{\sec^2\left(\frac{75^\circ}{2}\right)}} \]

### Step 4: Substitute \( \sec^2 \)
Using the identity \( \sec^2\theta = 1 + \tan^2\theta \):
\[ \sec^2\left(\frac{75^\circ}{2}\right) = 1 + \tan^2\left(\frac{75^\circ}{2}\right) \]
Thus, we can write:
\[ \frac{1}{1 - \frac{2}{1 + \tan^2\left(\frac{75^\circ}{2}\right)}} \]

### Step 5: Rationalize the Denominator
Now we can rationalize the denominator:
\[ 1 - \frac{2}{1 + \tan^2\left(\frac{75^\circ}{2}\right)} = \frac{(1 + \tan^2\left(\frac{75^\circ}{2}\right)) - 2}{1 + \tan^2\left(\frac{75^\circ}{2}\right)} = \frac{\tan^2\left(\frac{75^\circ}{2}\right) - 1}{1 + \tan^2\left(\frac{75^\circ}{2}\right)} \]

### Step 6: Final Expression
Thus, our expression simplifies to:
\[ -\frac{1}{\cos(75^\circ)} \]

### Step 7: Evaluate \( \cos(75^\circ) \)
Using the cosine addition formula:
\[ \cos(75^\circ) = \cos(45^\circ + 30^\circ) = \cos(45^\circ)\cos(30^\circ) - \sin(45^\circ)\sin(30^\circ) \]
Substituting known values:
\[ \cos(75^\circ) = \frac{1}{\sqrt{2}} \cdot \frac{\sqrt{3}}{2} - \frac{1}{\sqrt{2}} \cdot \frac{1}{2} = \frac{\sqrt{3} - 1}{2\sqrt{2}} \]

### Step 8: Final Value
Thus, the final value of the expression is:
\[ -\frac{2\sqrt{2}}{\sqrt{3} - 1} \]
After rationalizing, we get:
\[ -\sqrt{2}(\sqrt{3} + 1) \]

### Conclusion
The value of the expression is:
\[ -\sqrt{2}(\sqrt{3} + 1) \]

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