What is the value of `(sin (x-y))/(cos x cos y)+(sin (y-z))/(cos y cos z)+(sin (z-x))/(cos z cos x)` ?

Answer

To solve the expression

\[ \frac{\sin(x-y)}{\cos x \cos y} + \frac{\sin(y-z)}{\cos y \cos z} + \frac{\sin(z-x)}{\cos z \cos x} \]

we will use the sine subtraction formula and simplify each term step by step.

### Step 1: Simplify the first term

Using the sine subtraction formula, we have:

\[ \sin(x-y) = \sin x \cos y - \cos x \sin y \]

Thus, the first term becomes:

\[ \frac{\sin(x-y)}{\cos x \cos y} = \frac{\sin x \cos y - \cos x \sin y}{\cos x \cos y} \]

This can be split into two fractions:

\[ = \frac{\sin x \cos y}{\cos x \cos y} - \frac{\cos x \sin y}{\cos x \cos y} = \frac{\sin x}{\cos x} - \frac{\sin y}{\cos y} = \tan x - \tan y \]

### Step 2: Simplify the second term

Now, we simplify the second term:

\[ \frac{\sin(y-z)}{\cos y \cos z} = \frac{\sin y \cos z - \cos y \sin z}{\cos y \cos z} \]

Again, this can be split into two fractions:

\[ = \frac{\sin y \cos z}{\cos y \cos z} - \frac{\cos y \sin z}{\cos y \cos z} = \frac{\sin y}{\cos y} - \frac{\sin z}{\cos z} = \tan y - \tan z \]

### Step 3: Simplify the third term

Next, we simplify the third term:

\[ \frac{\sin(z-x)}{\cos z \cos x} = \frac{\sin z \cos x - \cos z \sin x}{\cos z \cos x} \]

This can also be split into two fractions:

\[ = \frac{\sin z \cos x}{\cos z \cos x} - \frac{\cos z \sin x}{\cos z \cos x} = \frac{\sin z}{\cos z} - \frac{\sin x}{\cos x} = \tan z - \tan x \]

### Step 4: Combine all terms

Now we combine all the simplified terms:

\[ (\tan x - \tan y) + (\tan y - \tan z) + (\tan z - \tan x) \]

When we combine these, we see that:

\[ \tan x - \tan y + \tan y - \tan z + \tan z - \tan x = 0 \]

### Final Answer

Thus, the value of the entire expression is:

\[ \boxed{0} \]