If ` f(x) = sin ( log x) ` then the value of ` E=f (xy) + f(x/y) - 2f (x) cos ( log y)` is

Options

A. `-1`
B. `0`
C. `1`
D. None of these

Answer

To solve the problem, we start with the function given:

\[ f(x) = \sin(\log x) \]

We need to find the value of:

\[ E = f(xy) + f\left(\frac{x}{y}\right) - 2f(x) \cos(\log y) \]

### Step 1: Calculate \( f(xy) \)

Using the definition of \( f(x) \):

\[ f(xy) = \sin(\log(xy)) = \sin(\log x + \log y) \]

### Step 2: Calculate \( f\left(\frac{x}{y}\right) \)

Using the definition of \( f(x) \):

\[ f\left(\frac{x}{y}\right) = \sin\left(\log\left(\frac{x}{y}\right)\right) = \sin(\log x - \log y) \]

### Step 3: Calculate \( f(x) \)

Using the definition of \( f(x) \):

\[ f(x) = \sin(\log x) \]

### Step 4: Substitute into \( E \)

Now we substitute these values into the expression for \( E \):

\[ E = \sin(\log x + \log y) + \sin(\log x - \log y) - 2\sin(\log x) \cos(\log y) \]

### Step 5: Use the sine addition and subtraction formulas

Using the sine addition and subtraction formulas:

\[ \sin(a + b) + \sin(a - b) = 2\sin(a)\cos(b) \]

Let \( a = \log x \) and \( b = \log y \):

\[ E = 2\sin(\log x)\cos(\log y) - 2\sin(\log x)\cos(\log y) \]

### Step 6: Simplify the expression

Now, we can see that:

\[ E = 2\sin(\log x)\cos(\log y) - 2\sin(\log x)\cos(\log y) = 0 \]

Thus, the value of \( E \) is:

\[ \boxed{0} \]