If the function `f(x)=(sqrt(1+cosx)+sqrt(1-cosx))/(sqrt(1+cosx)-sqrt(1-cosx))` If the value of `f(pi/3)=a+bsqrt(c)` then `a+b+c=`

Options

A. 4
B. 5
C. 6
D. 7

Answer

To solve the function \( f(x) = \frac{\sqrt{1 + \cos x} + \sqrt{1 - \cos x}}{\sqrt{1 + \cos x} - \sqrt{1 - \cos x}} \) and find \( f\left(\frac{\pi}{3}\right) \), we can follow these steps:

### Step 1: Substitute \( x = \frac{\pi}{3} \)
First, we need to calculate \( \cos\left(\frac{\pi}{3}\right) \):
\[ \cos\left(\frac{\pi}{3}\right) = \frac{1}{2} \]

Now substitute this value into the function:
\[ f\left(\frac{\pi}{3}\right) = \frac{\sqrt{1 + \frac{1}{2}} + \sqrt{1 - \frac{1}{2}}}{\sqrt{1 + \frac{1}{2}} - \sqrt{1 - \frac{1}{2}}} \]

### Step 2: Simplify the square roots
Calculate \( \sqrt{1 + \frac{1}{2}} \) and \( \sqrt{1 - \frac{1}{2}} \):
\[ \sqrt{1 + \frac{1}{2}} = \sqrt{\frac{3}{2}} = \frac{\sqrt{3}}{\sqrt{2}} = \frac{\sqrt{6}}{2} \]
\[ \sqrt{1 - \frac{1}{2}} = \sqrt{\frac{1}{2}} = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2} \]

### Step 3: Substitute back into the function
Now substitute these values back into the function:
\[ f\left(\frac{\pi}{3}\right) = \frac{\frac{\sqrt{6}}{2} + \frac{\sqrt{2}}{2}}{\frac{\sqrt{6}}{2} - \frac{\sqrt{2}}{2}} \]

### Step 4: Factor out \( \frac{1}{2} \)
Factor out \( \frac{1}{2} \) from both the numerator and denominator:
\[ f\left(\frac{\pi}{3}\right) = \frac{\frac{1}{2}(\sqrt{6} + \sqrt{2})}{\frac{1}{2}(\sqrt{6} - \sqrt{2})} = \frac{\sqrt{6} + \sqrt{2}}{\sqrt{6} - \sqrt{2}} \]

### Step 5: Rationalize the denominator
Multiply the numerator and denominator by the conjugate of the denominator:
\[ f\left(\frac{\pi}{3}\right) = \frac{(\sqrt{6} + \sqrt{2})(\sqrt{6} + \sqrt{2})}{(\sqrt{6} - \sqrt{2})(\sqrt{6} + \sqrt{2})} \]

### Step 6: Calculate the numerator and denominator
Calculate the numerator:
\[ (\sqrt{6} + \sqrt{2})^2 = 6 + 2 + 2\sqrt{12} = 8 + 4\sqrt{3} \]

Calculate the denominator:
\[ (\sqrt{6})^2 - (\sqrt{2})^2 = 6 - 2 = 4 \]

### Step 7: Final expression
Now we have:
\[ f\left(\frac{\pi}{3}\right) = \frac{8 + 4\sqrt{3}}{4} = 2 + \sqrt{3} \]

### Step 8: Identify \( a \), \( b \), and \( c \)
From the expression \( f\left(\frac{\pi}{3}\right) = 2 + 1\sqrt{3} \), we identify:
- \( a = 2 \)
- \( b = 1 \)
- \( c = 3 \)

### Step 9: Calculate \( a + b + c \)
Now calculate:
\[ a + b + c = 2 + 1 + 3 = 6 \]

Thus, the final answer is:
\[ \boxed{6} \]