To simplify the expression [tex]\((\sin(\theta) - \cos(\theta))^2\)[/tex], we will follow a step-by-step approach.
### Step 1: Expand using the Binomial Theorem
Recall that [tex]\((a - b)^2\)[/tex] can be expanded using the binomial theorem as:
[tex]\[ (a - b)^2 = a^2 - 2ab + b^2 \][/tex]
Here, [tex]\(a = \sin(\theta)\)[/tex] and [tex]\(b = \cos(\theta)\)[/tex]. Substituting these values into the binomial expansion, we get:
[tex]\[ (\sin(\theta) - \cos(\theta))^2 = \sin^2(\theta) - 2\sin(\theta)\cos(\theta) + \cos^2(\theta) \][/tex]
### Step 2: Use the Pythagorean Identity
There is a well-known trigonometric identity, called the Pythagorean identity, which states:
[tex]\[ \sin^2(\theta) + \cos^2(\theta) = 1 \][/tex]
Using this identity, we can substitute the sum [tex]\(\sin^2(\theta) + \cos^2(\theta)\)[/tex] with [tex]\(1\)[/tex].
### Step 3: Substitute and Simplify
Substitute the Pythagorean identity into our expanded expression:
[tex]\[ \sin^2(\theta) - 2\sin(\theta)\cos(\theta) + \cos^2(\theta) \][/tex]
[tex]\[ = (\sin^2(\theta) + \cos^2(\theta)) - 2\sin(\theta)\cos(\theta) \][/tex]
[tex]\[ = 1 - 2\sin(\theta)\cos(\theta) \][/tex]
### Final Simplified Expression
Therefore, the simplified form of the given expression is:
[tex]\[ (\sin(\theta) - \cos(\theta))^2 = 1 - 2\sin(\theta)\cos(\theta) \][/tex]
So the final answer is:
[tex]\[ 1 - 2\sin(\theta)\cos(\theta) \][/tex]
