`(1-cos x)/(cos x(1+cos x ))=(1)/(cos x)-(2)/(1+cos x)`

Answer

To solve the equation

\[ \frac{1 - \cos x}{\cos x(1 + \cos x)} = \frac{1}{\cos x} - \frac{2}{1 + \cos x} \]

we will start by simplifying both sides and checking if they are equal.

### Step 1: Rewrite the left-hand side

The left-hand side is

\[ \frac{1 - \cos x}{\cos x(1 + \cos x)} \]

### Step 2: Rewrite the right-hand side

The right-hand side is

\[ \frac{1}{\cos x} - \frac{2}{1 + \cos x} \]

To combine these fractions, we need a common denominator, which is \(\cos x(1 + \cos x)\).

### Step 3: Combine the right-hand side

Rewriting the right-hand side with a common denominator:

\[ \frac{1}{\cos x} = \frac{1(1 + \cos x)}{\cos x(1 + \cos x)} = \frac{1 + \cos x}{\cos x(1 + \cos x)} \]

Now, rewrite the second term:

\[ -\frac{2}{1 + \cos x} = -\frac{2\cos x}{\cos x(1 + \cos x)} \]

Now, we can combine both fractions:

\[ \frac{1 + \cos x - 2\cos x}{\cos x(1 + \cos x)} = \frac{1 - \cos x}{\cos x(1 + \cos x)} \]

### Step 4: Compare both sides

Now we have:

Left-hand side:

\[ \frac{1 - \cos x}{\cos x(1 + \cos x)} \]

Right-hand side:

\[ \frac{1 - \cos x}{\cos x(1 + \cos x)} \]

Since both sides are equal, we conclude that the given statement is true.

### Conclusion

Thus, we have shown that

\[ \frac{1 - \cos x}{\cos x(1 + \cos x)} = \frac{1}{\cos x} - \frac{2}{1 + \cos x} \]

is indeed true.

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