`(cot^(2)15^(@)-1)/(cot^(2)15^(@)+1)=?`

Options

A. `(sqrt(3))/(2)`
B. `(2)/(sqrt(3))`
C. 1
D. 0

Answer

To solve the expression \((\cot^2 15^\circ - 1)/(\cot^2 15^\circ + 1)\), we can follow these steps:

### Step 1: Rewrite cotangent in terms of tangent
We know that \(\cot \theta = \frac{1}{\tan \theta}\). Therefore, we can express \(\cot^2 15^\circ\) as:
\[ \cot^2 15^\circ = \frac{1}{\tan^2 15^\circ} \]

### Step 2: Substitute cotangent in the expression
Substituting \(\cot^2 15^\circ\) into the expression gives us:
\[ \frac{\frac{1}{\tan^2 15^\circ} - 1}{\frac{1}{\tan^2 15^\circ} + 1} \]

### Step 3: Simplify the expression
To simplify, we can multiply the numerator and the denominator by \(\tan^2 15^\circ\):
\[ \frac{1 - \tan^2 15^\circ}{1 + \tan^2 15^\circ} \]

### Step 4: Use the double angle identity for cosine
We can use the identity:
\[ \cos 2\theta = \frac{1 - \tan^2 \theta}{1 + \tan^2 \theta} \]
Setting \(\theta = 15^\circ\), we have:
\[ \cos 30^\circ = \frac{1 - \tan^2 15^\circ}{1 + \tan^2 15^\circ} \]

### Step 5: Calculate \(\cos 30^\circ\)
We know that:
\[ \cos 30^\circ = \frac{\sqrt{3}}{2} \]

### Final Result
Thus, the value of the original expression is:
\[ \frac{\cot^2 15^\circ - 1}{\cot^2 15^\circ + 1} = \cos 30^\circ = \frac{\sqrt{3}}{2} \]

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