The value of `tan63^(@)-cot63^(@)` is equal to

Options

A. `(sqrt5-1)/(sqrt5+1)sqrt(10+2sqrt5)`
B. `(2)/(sqrt5+1)sqrt(10+2sqrt5)`
C. `(sqrt5-1)/(4)sqrt(10-2sqrt5)`
D. `(sqrt5-1)/(4)sqrt(10+2sqrt5)`

Answer

To find the value of \( \tan 63^\circ - \cot 63^\circ \), we can follow these steps:

### Step 1: Rewrite the Trigonometric Functions
We start by expressing \( \tan \) and \( \cot \) in terms of sine and cosine:
\[ \tan 63^\circ = \frac{\sin 63^\circ}{\cos 63^\circ} \]
\[ \cot 63^\circ = \frac{\cos 63^\circ}{\sin 63^\circ} \]

### Step 2: Find a Common Denominator
Now, we can combine these two fractions:
\[ \tan 63^\circ - \cot 63^\circ = \frac{\sin 63^\circ}{\cos 63^\circ} - \frac{\cos 63^\circ}{\sin 63^\circ} \]
To combine these, we find a common denominator, which is \( \cos 63^\circ \sin 63^\circ \):
\[ = \frac{\sin^2 63^\circ - \cos^2 63^\circ}{\cos 63^\circ \sin 63^\circ} \]

### Step 3: Apply the Difference of Squares
Notice that \( \sin^2 63^\circ - \cos^2 63^\circ \) can be rewritten using the identity:
\[ \sin^2 \theta - \cos^2 \theta = -\cos(2\theta) \]
Thus, we have:
\[ \sin^2 63^\circ - \cos^2 63^\circ = -\cos(126^\circ) \]
So, substituting this back, we get:
\[ \tan 63^\circ - \cot 63^\circ = \frac{-\cos(126^\circ)}{\cos 63^\circ \sin 63^\circ} \]

### Step 4: Simplify Further
Using the identity \( \cos(126^\circ) = -\cos(54^\circ) \) (since \( 126^\circ = 180^\circ - 54^\circ \)), we can rewrite:
\[ = \frac{\cos(54^\circ)}{\cos 63^\circ \sin 63^\circ} \]

### Step 5: Use Known Values
We know that:
\[ \sin 63^\circ = \cos 27^\circ \quad \text{and} \quad \cos 63^\circ = \sin 27^\circ \]
Thus, we can express:
\[ \sin 63^\circ \cos 63^\circ = \sin 27^\circ \cos 27^\circ = \frac{1}{2} \sin(54^\circ) \]
So, we can substitute this back into our expression:
\[ \tan 63^\circ - \cot 63^\circ = \frac{\cos(54^\circ)}{\frac{1}{2} \sin(54^\circ)} = \frac{2 \cos(54^\circ)}{\sin(54^\circ)} = 2 \cot(54^\circ) \]

### Step 6: Final Simplification
Since \( \cot(54^\circ) = \tan(36^\circ) \), we can write:
\[ \tan 63^\circ - \cot 63^\circ = 2 \tan(36^\circ) \]

### Conclusion
Therefore, the final value of \( \tan 63^\circ - \cot 63^\circ \) is:
\[ \boxed{2 \tan(36^\circ)} \]