Certainly! Let's solve the given expression step-by-step:
Given expression:
[tex]\[ (\sin x + \cos x)^2 - (\sin x + 2 \sin x \cos x) \][/tex]
### Step 1: Expand the first part
First, we expand [tex]\((\sin x + \cos x)^2\)[/tex].
[tex]\[ (\sin x + \cos x)^2 = (\sin x + \cos x)(\sin x + \cos x) \][/tex]
Using the distributive property (FOIL method):
[tex]\[ (\sin x + \cos x)^2 = \sin^2 x + \sin x \cos x + \cos x \sin x + \cos^2 x \][/tex]
[tex]\[ = \sin^2 x + 2 \sin x \cos x + \cos^2 x \][/tex]
### Step 2: Simplify the First Part
Recall the Pythagorean identity: [tex]\(\sin^2 x + \cos^2 x = 1\)[/tex].
So,
[tex]\[ \sin^2 x + 2 \sin x \cos x + \cos^2 x = 1 + 2 \sin x \cos x \][/tex]
### Step 3: Rewrite the expression
Our expression now becomes:
[tex]\[ 1 + 2 \sin x \cos x - (\sin x + 2 \sin x \cos x) \][/tex]
### Step 4: Distribute the negative sign
Distribute the negative sign through the parentheses:
[tex]\[ 1 + 2 \sin x \cos x - \sin x - 2 \sin x \cos x \][/tex]
### Step 5: Combine like terms
Notice that [tex]\(2 \sin x \cos x\)[/tex] and [tex]\(-2 \sin x \cos x\)[/tex] cancel each other out:
[tex]\[ 1 - \sin x \][/tex]
### Final Simplified Expression:
The final expression simplifies to:
[tex]\[ 1 - \sin x \][/tex]
So, the step-by-step solution to the given expression [tex]\( (\sin x + \cos x)^2 - (\sin x + 2 \sin x \cos x) \)[/tex] is:
[tex]\[ 1 - \sin x \][/tex]
