Answer :
Let's simplify the given expression step by step:
[tex]\[ \frac{1 - 2 \sin x \cos x}{(\sin x + \cos x)(\sin x - \cos x)} \][/tex]
### Step 1: Expand the Denominator
First, we expand the denominator [tex]\((\sin x + \cos x)(\sin x - \cos x)\)[/tex] using the difference of squares formula:
[tex]\[ (\sin x + \cos x)(\sin x - \cos x) = \sin^2 x - \cos^2 x \][/tex]
### Step 2: Substitute Back
Rewrite the fraction with the expanded denominator:
[tex]\[ \frac{1 - 2 \sin x \cos x}{\sin^2 x - \cos^2 x} \][/tex]
### Step 3: Recognize Trigonometric Identities
Note that [tex]\( \sin(2x) = 2 \sin x \cos x \)[/tex] and recall the Pythagorean identity-related expression for [tex]\(\cos(2x)\)[/tex]:
[tex]\[ \cos(2x) = \cos^2 x - \sin^2 x \][/tex]
### Step 4: Substitute and Simplify
We can use these identities to rewrite the fraction. Rewriting [tex]\(\cos^2 x - \sin^2 x\)[/tex] as [tex]\(-(\sin^2 x - \cos^2 x)\)[/tex]:
[tex]\[ \sin^2 x - \cos^2 x = -(\cos^2 x - \sin^2 x) = -\cos(2x) \][/tex]
So, our expression becomes:
[tex]\[ \frac{1 - 2 \sin x \cos x}{-\cos(2x)} \][/tex]
Now substitute [tex]\(\sin(2x)\)[/tex] for [tex]\(2 \sin x \cos x\)[/tex]:
[tex]\[ 1 - 2 \sin x \cos x = 1 - \sin(2x) \][/tex]
Thus, the numerator becomes:
[tex]\[ 1 - \sin(2x) \][/tex]
### Final Simplified Form
Putting it all together, we get:
[tex]\[ \frac{1 - \sin(2x)}{-\cos(2x)} = \frac{\sin(2x) - 1}{\cos(2x)} \][/tex]
Thus, the simplified expression is:
[tex]\[ \frac{\sin(2x) - 1}{\cos(2x)} \][/tex]
Therefore, the solution to the given expression is:
[tex]\[ \frac{1 - 2 \sin x \cos x}{(\sin x + \cos x)(\sin x - \cos x)} = \frac{\sin(2x) - 1}{\cos(2x)} \][/tex]
[tex]\[ \frac{1 - 2 \sin x \cos x}{(\sin x + \cos x)(\sin x - \cos x)} \][/tex]
### Step 1: Expand the Denominator
First, we expand the denominator [tex]\((\sin x + \cos x)(\sin x - \cos x)\)[/tex] using the difference of squares formula:
[tex]\[ (\sin x + \cos x)(\sin x - \cos x) = \sin^2 x - \cos^2 x \][/tex]
### Step 2: Substitute Back
Rewrite the fraction with the expanded denominator:
[tex]\[ \frac{1 - 2 \sin x \cos x}{\sin^2 x - \cos^2 x} \][/tex]
### Step 3: Recognize Trigonometric Identities
Note that [tex]\( \sin(2x) = 2 \sin x \cos x \)[/tex] and recall the Pythagorean identity-related expression for [tex]\(\cos(2x)\)[/tex]:
[tex]\[ \cos(2x) = \cos^2 x - \sin^2 x \][/tex]
### Step 4: Substitute and Simplify
We can use these identities to rewrite the fraction. Rewriting [tex]\(\cos^2 x - \sin^2 x\)[/tex] as [tex]\(-(\sin^2 x - \cos^2 x)\)[/tex]:
[tex]\[ \sin^2 x - \cos^2 x = -(\cos^2 x - \sin^2 x) = -\cos(2x) \][/tex]
So, our expression becomes:
[tex]\[ \frac{1 - 2 \sin x \cos x}{-\cos(2x)} \][/tex]
Now substitute [tex]\(\sin(2x)\)[/tex] for [tex]\(2 \sin x \cos x\)[/tex]:
[tex]\[ 1 - 2 \sin x \cos x = 1 - \sin(2x) \][/tex]
Thus, the numerator becomes:
[tex]\[ 1 - \sin(2x) \][/tex]
### Final Simplified Form
Putting it all together, we get:
[tex]\[ \frac{1 - \sin(2x)}{-\cos(2x)} = \frac{\sin(2x) - 1}{\cos(2x)} \][/tex]
Thus, the simplified expression is:
[tex]\[ \frac{\sin(2x) - 1}{\cos(2x)} \][/tex]
Therefore, the solution to the given expression is:
[tex]\[ \frac{1 - 2 \sin x \cos x}{(\sin x + \cos x)(\sin x - \cos x)} = \frac{\sin(2x) - 1}{\cos(2x)} \][/tex]
