FInd : `tan75^0+cot75^0=`

Options

A. 1
B. 2
C. 3
D. 4

Explanation

To solve the problem \( \tan 75^\circ + \cot 75^\circ \), we will follow these steps:

### Step 1: Express \( \tan 75^\circ \) using the angle addition formula
We can express \( 75^\circ \) as \( 45^\circ + 30^\circ \). The angle addition formula for tangent is given by:

\[ \tan(a + b) = \frac{\tan a + \tan b}{1 - \tan a \tan b} \]

Here, let \( a = 45^\circ \) and \( b = 30^\circ \). Therefore,

\[ \tan 75^\circ = \tan(45^\circ + 30^\circ) = \frac{\tan 45^\circ + \tan 30^\circ}{1 - \tan 45^\circ \tan 30^\circ} \]

### Step 2: Substitute the known values of \( \tan 45^\circ \) and \( \tan 30^\circ \)
We know that:

\[ \tan 45^\circ = 1 \quad \text{and} \quad \tan 30^\circ = \frac{1}{\sqrt{3}} \]

Substituting these values into the formula gives:

\[ \tan 75^\circ = \frac{1 + \frac{1}{\sqrt{3}}}{1 - 1 \cdot \frac{1}{\sqrt{3}}} \]

### Step 3: Simplify the expression for \( \tan 75^\circ \)
Now, simplify the numerator and denominator:

Numerator:
\[ 1 + \frac{1}{\sqrt{3}} = \frac{\sqrt{3} + 1}{\sqrt{3}} \]

Denominator:
\[ 1 - \frac{1}{\sqrt{3}} = \frac{\sqrt{3} - 1}{\sqrt{3}} \]

Thus, we have:

\[ \tan 75^\circ = \frac{\frac{\sqrt{3} + 1}{\sqrt{3}}}{\frac{\sqrt{3} - 1}{\sqrt{3}}} = \frac{\sqrt{3} + 1}{\sqrt{3} - 1} \]

### Step 4: Find \( \cot 75^\circ \)
We know that \( \cot \theta = \frac{1}{\tan \theta} \). Therefore,

\[ \cot 75^\circ = \frac{1}{\tan 75^\circ} = \frac{\sqrt{3} - 1}{\sqrt{3} + 1} \]

### Step 5: Add \( \tan 75^\circ \) and \( \cot 75^\circ \)
Now we can add \( \tan 75^\circ \) and \( \cot 75^\circ \):

\[ \tan 75^\circ + \cot 75^\circ = \frac{\sqrt{3} + 1}{\sqrt{3} - 1} + \frac{\sqrt{3} - 1}{\sqrt{3} + 1} \]

### Step 6: Simplify the sum
To add these fractions, we need 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: Calculate the numerator
Calculating the squares:

\[ (\sqrt{3} + 1)^2 = 3 + 2\sqrt{3} + 1 = 4 + 2\sqrt{3} \]
\[ (\sqrt{3} - 1)^2 = 3 - 2\sqrt{3} + 1 = 4 - 2\sqrt{3} \]

Adding these:

\[ (4 + 2\sqrt{3}) + (4 - 2\sqrt{3}) = 8 \]

### Step 8: Calculate the denominator
The denominator simplifies to:

\[ (\sqrt{3} - 1)(\sqrt{3} + 1) = 3 - 1 = 2 \]

### Step 9: Final result
Thus, we have:

\[ \tan 75^\circ + \cot 75^\circ = \frac{8}{2} = 4 \]

### Conclusion
Therefore, the final result is:

\[ \tan 75^\circ + \cot 75^\circ = 4 \]

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