The greatest value of the term independent of x, as `alpha` varies over R, in the expansion of `(xcosalpha+(sinalpha)/x)^(20)` is

Options

A. `""^(20)C_(10)`
B. `""^(20)C_(15)`
C. `""^(20)C_(19)`
D. `(""^(20)C_(10))/(2^10)`

Answer

To find the greatest value of the term independent of \( x \) in the expansion of \( (x \cos \alpha + \frac{\sin \alpha}{x})^{20} \), we can follow these steps:

### Step 1: Identify the general term in the binomial expansion
The general term \( T_r \) in the expansion of \( (x \cos \alpha + \frac{\sin \alpha}{x})^{20} \) can be expressed as:
\[ T_r = \binom{20}{r} (x \cos \alpha)^r \left(\frac{\sin \alpha}{x}\right)^{20 - r} \]
This simplifies to:
\[ T_r = \binom{20}{r} (\cos \alpha)^r (\sin \alpha)^{20 - r} x^{r - (20 - r)} = \binom{20}{r} (\cos \alpha)^r (\sin \alpha)^{20 - r} x^{2r - 20} \]

### Step 2: Find the term independent of \( x \)
To find the term that is independent of \( x \), we need to set the exponent of \( x \) to zero:
\[ 2r - 20 = 0 \implies r = 10 \]
Thus, the term independent of \( x \) is \( T_{10} \).

### Step 3: Calculate \( T_{10} \)
Substituting \( r = 10 \) into the expression for \( T_r \):
\[ T_{10} = \binom{20}{10} (\cos \alpha)^{10} (\sin \alpha)^{10} \]
This can be rewritten using the identity \( \sin 2\alpha = 2 \sin \alpha \cos \alpha \):
\[ T_{10} = \binom{20}{10} \left(\frac{\sin 2\alpha}{2}\right)^{10} \]
Thus,
\[ T_{10} = \binom{20}{10} \frac{(\sin 2\alpha)^{10}}{2^{10}} \]

### Step 4: Maximize \( T_{10} \)
To find the maximum value of \( T_{10} \), we need to maximize \( (\sin 2\alpha)^{10} \). The maximum value of \( \sin 2\alpha \) is 1, which occurs when \( 2\alpha = \frac{\pi}{2} + k\pi \) for \( k \in \mathbb{Z} \). Thus, the maximum value of \( (\sin 2\alpha)^{10} \) is \( 1^{10} = 1 \).

### Step 5: Final expression for the maximum value
Substituting this back into our expression for \( T_{10} \):
\[ \text{Max value of } T_{10} = \binom{20}{10} \frac{1}{2^{10}} \]

### Step 6: Calculate \( \binom{20}{10} \)
Using the formula for combinations:
\[ \binom{20}{10} = \frac{20!}{10! \cdot 10!} \]
Thus, the maximum value of the term independent of \( x \) is:
\[ \text{Max value} = \frac{20!}{10! \cdot 10!} \cdot \frac{1}{2^{10}} \]

### Conclusion
The greatest value of the term independent of \( x \) in the expansion of \( (x \cos \alpha + \frac{\sin \alpha}{x})^{20} \) is:
\[ \frac{20!}{10! \cdot 10! \cdot 2^{10}} \]