Value of tan `75^(@) + cot 75^(@) = ? `

Options

A. `(1)/(4)`
B. `sqrt(3)`
C. 4
D. `-4`

Answer

To find the value of \( \tan 75^\circ + \cot 75^\circ \), we will follow these steps:

### Step 1: Express \( \tan 75^\circ \)
We can express \( \tan 75^\circ \) as:
\[ \tan 75^\circ = \tan(45^\circ + 30^\circ) \]
Using the formula for the tangent of a sum, we have:
\[ \tan(a + b) = \frac{\tan a + \tan b}{1 - \tan a \tan b} \]
Substituting \( a = 45^\circ \) and \( b = 30^\circ \):
\[ \tan 75^\circ = \frac{\tan 45^\circ + \tan 30^\circ}{1 - \tan 45^\circ \tan 30^\circ} \]

### Step 2: Substitute known values
We know that:
\[ \tan 45^\circ = 1 \quad \text{and} \quad \tan 30^\circ = \frac{1}{\sqrt{3}} \]
Substituting these values into the equation:
\[ \tan 75^\circ = \frac{1 + \frac{1}{\sqrt{3}}}{1 - 1 \cdot \frac{1}{\sqrt{3}}} \]

### Step 3: Simplify the expression
Now we simplify the numerator and denominator:
\[ \tan 75^\circ = \frac{1 + \frac{1}{\sqrt{3}}}{1 - \frac{1}{\sqrt{3}}} \]
To simplify, multiply the numerator and denominator by \( \sqrt{3} \):
\[ \tan 75^\circ = \frac{\sqrt{3} + 1}{\sqrt{3} - 1} \]

### Step 4: Find \( \cot 75^\circ \)
Since \( \cot 75^\circ = \frac{1}{\tan 75^\circ} \):
\[ \cot 75^\circ = \frac{\sqrt{3} - 1}{\sqrt{3} + 1} \]

### Step 5: Add \( \tan 75^\circ \) and \( \cot 75^\circ \)
Now we can find \( \tan 75^\circ + \cot 75^\circ \):
\[ \tan 75^\circ + \cot 75^\circ = \frac{\sqrt{3} + 1}{\sqrt{3} - 1} + \frac{\sqrt{3} - 1}{\sqrt{3} + 1} \]

### Step 6: Combine the fractions
To combine the fractions, we find a common denominator:
\[ \tan 75^\circ + \cot 75^\circ = \frac{(\sqrt{3} + 1)^2 + (\sqrt{3} - 1)^2}{(\sqrt{3} - 1)(\sqrt{3} + 1)} \]

### Step 7: Simplify the numerator and denominator
Calculating the numerator:
\[ (\sqrt{3} + 1)^2 = 3 + 2\sqrt{3} + 1 = 4 + 2\sqrt{3} \]
\[ (\sqrt{3} - 1)^2 = 3 - 2\sqrt{3} + 1 = 4 - 2\sqrt{3} \]
Adding these:
\[ 4 + 2\sqrt{3} + 4 - 2\sqrt{3} = 8 \]
Calculating the denominator:
\[ (\sqrt{3} - 1)(\sqrt{3} + 1) = 3 - 1 = 2 \]

### Step 8: Final result
Thus, we have:
\[ \tan 75^\circ + \cot 75^\circ = \frac{8}{2} = 4 \]

### Final Answer
The value of \( \tan 75^\circ + \cot 75^\circ \) is \( 4 \).