Find the value: `1 + tan15^@tan75^@`

Options

A. `sec^2 15^@`
B. `sec^2 75^@`
C. 2
D. `1+(1+sqrt3)/2`

Answer

To solve the problem \(1 + \tan 15^\circ \tan 75^\circ\), we can follow these steps:

### Step 1: Use the Identity for Tangent
Recall that \(\tan(90^\circ - \theta) = \cot(\theta)\). Therefore, we can express \(\tan 75^\circ\) as:
\[ \tan 75^\circ = \tan(90^\circ - 15^\circ) = \cot 15^\circ \]

### Step 2: Substitute in the Expression
Now substitute \(\tan 75^\circ\) in the original expression:
\[ 1 + \tan 15^\circ \tan 75^\circ = 1 + \tan 15^\circ \cot 15^\circ \]

### Step 3: Simplify Using Cotangent
We know that \(\cot \theta = \frac{1}{\tan \theta}\). Thus, we can write:
\[ \tan 15^\circ \cot 15^\circ = \tan 15^\circ \cdot \frac{1}{\tan 15^\circ} = 1 \]

### Step 4: Final Calculation
Now substitute this back into the expression:
\[ 1 + \tan 15^\circ \cot 15^\circ = 1 + 1 = 2 \]

### Conclusion
The final value is:
\[ \boxed{2} \]