The value of tan75

Options

A. 10
B. 1
C. 2
D. 20

Answer

To find the value of \( \tan 75^\circ \), we can use the angle addition formula for tangent. Here’s a step-by-step solution:

### Step 1: Express \( 75^\circ \) as a sum of two angles
We can express \( 75^\circ \) as:
\[ 75^\circ = 45^\circ + 30^\circ \]

### Step 2: Use the tangent addition formula
The tangent addition formula states that:
\[ \tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} \]
In our case, let \( A = 45^\circ \) and \( B = 30^\circ \). Thus, we have:
\[ \tan 75^\circ = \tan(45^\circ + 30^\circ) = \frac{\tan 45^\circ + \tan 30^\circ}{1 - \tan 45^\circ \tan 30^\circ} \]

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

### Step 4: Simplify the expression
Now, simplify the numerator and the denominator:
- **Numerator**:
\[ 1 + \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{\sqrt{3}} + \frac{1}{\sqrt{3}} = \frac{\sqrt{3} + 1}{\sqrt{3}} \]

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

Putting it all together:
\[ \tan 75^\circ = \frac{\frac{\sqrt{3} + 1}{\sqrt{3}}}{\frac{\sqrt{3} - 1}{\sqrt{3}}} \]

### Step 5: Cancel out \( \sqrt{3} \)
Since both the numerator and the denominator have \( \sqrt{3} \) in them, we can cancel it out:
\[ \tan 75^\circ = \frac{\sqrt{3} + 1}{\sqrt{3} - 1} \]

### Final Result
Thus, the value of \( \tan 75^\circ \) is:
\[ \tan 75^\circ = \frac{\sqrt{3} + 1}{\sqrt{3} - 1} \]