The value of `(cot^(n)29^(@)-cot^(n)61^(@))/(tan^(n)61^(@)-tan^(n)29^(@))=?`

Options

A. 5
B. 7
C. 1
D. 3

Answer

To solve the problem \((\cot^n 29^\circ - \cot^n 61^\circ) / (\tan^n 61^\circ - \tan^n 29^\circ)\), we can follow these steps:

### Step 1: Rewrite the cotangent and tangent functions
We know that:
\[ \cot 61^\circ = \tan (90^\circ - 61^\circ) = \tan 29^\circ \]
This means that:
\[ \cot 29^\circ = \frac{1}{\tan 29^\circ} \]
and
\[ \cot 61^\circ = \frac{1}{\tan 61^\circ} \]

### Step 2: Substitute cotangent values
Substituting these into the original expression gives:
\[ \frac{\left(\frac{1}{\tan 29^\circ}\right)^n - \left(\frac{1}{\tan 61^\circ}\right)^n}{\tan^n 61^\circ - \tan^n 29^\circ} \]

### Step 3: Simplify the numerator
The numerator can be rewritten as:
\[ \frac{1^n - \left(\frac{\tan 29^\circ}{\tan 61^\circ}\right)^n}{\tan^n 61^\circ - \tan^n 29^\circ} \]
This simplifies to:
\[ \frac{1 - \left(\frac{\tan 29^\circ}{\tan 61^\circ}\right)^n}{\tan^n 61^\circ - \tan^n 29^\circ} \]

### Step 4: Factor the denominator
The denominator can be factored using the difference of powers:
\[ \tan^n 61^\circ - \tan^n 29^\circ = (\tan 61^\circ - \tan 29^\circ)(\tan^{n-1} 61^\circ + \tan^{n-1} 29^\circ + \cdots + \tan^{n-1} 29^\circ) \]

### Step 5: Substitute back into the expression
Now substituting this back gives:
\[ \frac{1 - \left(\frac{\tan 29^\circ}{\tan 61^\circ}\right)^n}{(\tan 61^\circ - \tan 29^\circ)(\tan^{n-1} 61^\circ + \tan^{n-1} 29^\circ + \cdots + \tan^{n-1} 29^\circ)} \]

### Step 6: Evaluate the expression
Notice that since \(\tan 61^\circ = \cot 29^\circ\), we can simplify the expression further. After simplification, we find that the numerator and denominator will cancel out, leading to:
\[ 1 \]

### Final Answer
Thus, the value of the expression is:
\[ \boxed{1} \]