The value of `cot7(1^0)/2+tan67(1^(@))/2-cot67(1^(@))/2-tan7(1^0)/2` is

Options

A. a rational number
B. irrational number
C. `2(3+2sqrt(3))`
D. `2(3-sqrt(3))`

Answer

To solve the problem, we need to evaluate the expression:

\[ \cot \left( 7 \frac{1}{2} \right) + \tan \left( 67 \frac{1}{2} \right) - \cot \left( 67 \frac{1}{2} \right) - \tan \left( 7 \frac{1}{2} \right) \]

### Step 1: Convert Mixed Numbers to Improper Fractions
First, we convert the mixed fractions into improper fractions:
- \( 7 \frac{1}{2} = \frac{15}{2} \)
- \( 67 \frac{1}{2} = \frac{135}{2} \)

So, the expression becomes:

\[ \cot \left( \frac{15}{2} \right) + \tan \left( \frac{135}{2} \right) - \cot \left( \frac{135}{2} \right) - \tan \left( \frac{15}{2} \right) \]

### Step 2: Rewrite Cotangent and Tangent in Terms of Sine and Cosine
Using the definitions of cotangent and tangent, we rewrite the expression:

\[ \cot \left( \frac{15}{2} \right) = \frac{\cos \left( \frac{15}{2} \right)}{\sin \left( \frac{15}{2} \right)}, \quad \tan \left( \frac{135}{2} \right) = \frac{\sin \left( \frac{135}{2} \right)}{\cos \left( \frac{135}{2} \right)} \]

Thus, the expression becomes:

\[ \frac{\cos \left( \frac{15}{2} \right)}{\sin \left( \frac{15}{2} \right)} + \frac{\sin \left( \frac{135}{2} \right)}{\cos \left( \frac{135}{2} \right)} - \frac{\cos \left( \frac{135}{2} \right)}{\sin \left( \frac{135}{2} \right)} - \frac{\sin \left( \frac{15}{2} \right)}{\cos \left( \frac{15}{2} \right)} \]

### Step 3: Combine Terms
Now, we can combine the terms by finding a common denominator. The common denominator for each pair of terms is \( \sin \left( \frac{15}{2} \right) \cos \left( \frac{15}{2} \right) \) and \( \sin \left( \frac{135}{2} \right) \cos \left( \frac{135}{2} \right) \).

This results in:

\[ \frac{\cos^2 \left( \frac{15}{2} \right) - \sin^2 \left( \frac{15}{2} \right)}{\sin \left( \frac{15}{2} \right) \cos \left( \frac{15}{2} \right)} + \frac{\sin^2 \left( \frac{135}{2} \right) - \cos^2 \left( \frac{135}{2} \right)}{\sin \left( \frac{135}{2} \right) \cos \left( \frac{135}{2} \right)} \]

### Step 4: Use Trigonometric Identities
Using the identities:
- \( \cos(2\theta) = \cos^2(\theta) - \sin^2(\theta) \)
- \( \sin(2\theta) = 2 \sin(\theta) \cos(\theta) \)

We rewrite the expression:

\[ \frac{2 \cos \left( 15 \right)}{\sin \left( 15 \right)} - \frac{2 \cos \left( 135 \right)}{\sin \left( 135 \right)} \]

### Step 5: Evaluate Cotangent Values
We know that \( \cot(135) = -1 \), so we can substitute:

\[ = 2 \left( \cot(15) + 1 \right) \]

### Step 6: Find the Value of \( \cot(15) \)
Using the angle subtraction formula:

\[ \tan(15) = \tan(45 - 30) = \frac{\tan(45) - \tan(30)}{1 + \tan(45) \tan(30)} = \frac{1 - \frac{1}{\sqrt{3}}}{1 + \frac{1}{\sqrt{3}}} \]

### Step 7: Rationalize and Simplify
After rationalizing, we find:

\[ \tan(15) = \frac{\sqrt{3} - 1}{\sqrt{3} + 1} \]

Thus,

\[ \cot(15) = \frac{1}{\tan(15)} = \frac{\sqrt{3} + 1}{\sqrt{3} - 1} \]

### Step 8: Final Calculation
Substituting back into our expression gives us:

\[ 2 \left( \frac{\sqrt{3} + 1}{\sqrt{3} - 1} + 1 \right) \]

### Conclusion
After simplification, we find the final value of the original expression.