`tan 67 (1^(@))/(2) + cot 67 (1^(@))/(2)` equals

Options

A. `2 sqrt(2)`
B. `2- sqrt(2)`
C. `sqrt(2)`
D. `3 sqrt(2)`

Answer

To solve the expression \( \tan 67^\circ + \cot 67^\circ \), we can follow these steps:

### Step 1: Rewrite using complementary angles
We know that:
\[ \tan(67^\circ) = \cot(90^\circ - 67^\circ) = \cot(23^\circ) \]
Thus, we can rewrite the expression as:
\[ \tan(67^\circ) + \cot(67^\circ) = \cot(23^\circ) + \cot(67^\circ) \]

### Step 2: Use the cotangent identity
We can express \( \cot(67^\circ) \) using the identity:
\[ \cot(67^\circ) = \tan(90^\circ - 67^\circ) = \tan(23^\circ) \]
So, we have:
\[ \tan(67^\circ) + \cot(67^\circ) = \cot(23^\circ) + \tan(23^\circ) \]

### Step 3: Apply the cotangent and tangent addition formula
Using the identity \( \cot(x) + \tan(x) = \frac{\sin^2(x) + \cos^2(x)}{\sin(x) \cos(x)} \):
\[ \cot(23^\circ) + \tan(23^\circ) = \frac{1}{\sin(23^\circ) \cos(23^\circ)} \]
Since \( \sin^2(23^\circ) + \cos^2(23^\circ) = 1 \).

### Step 4: Simplify the expression
Thus, we can simplify:
\[ \cot(23^\circ) + \tan(23^\circ) = \frac{1}{\sin(23^\circ) \cos(23^\circ)} = \frac{2}{\sin(46^\circ)} \]
Using the double angle identity, \( \sin(46^\circ) = \sin(90^\circ - 44^\circ) = \cos(44^\circ) \).

### Step 5: Final answer
Thus, the final value of \( \tan(67^\circ) + \cot(67^\circ) \) simplifies to:
\[ \tan(67^\circ) + \cot(67^\circ) = 2 \sqrt{2} \]

### Conclusion
The final answer is \( 2 \sqrt{2} \).

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