Answer :
Sure, let's simplify the given expression [tex]\((\sin(\theta) - \cos(\theta))^2\)[/tex].
Here’s the step-by-step solution:
1. Expand the Expression:
[tex]\[ (\sin(\theta) - \cos(\theta))^2 \][/tex]
We use the algebraic identity for the square of a binomial: [tex]\((a - b)^2 = a^2 - 2ab + b^2\)[/tex]. In this case, [tex]\(a = \sin(\theta)\)[/tex] and [tex]\(b = \cos(\theta)\)[/tex].
2. Apply the Identity:
[tex]\[ (\sin(\theta) - \cos(\theta))^2 = \sin(\theta)^2 - 2\sin(\theta)\cos(\theta) + \cos(\theta)^2 \][/tex]
3. Combine Like Terms:
[tex]\[ \sin(\theta)^2 - 2\sin(\theta)\cos(\theta) + \cos(\theta)^2 \][/tex]
So, the simplified expression is:
[tex]\[ \sin(\theta)^2 - 2\sin(\theta)\cos(\theta) + \cos(\theta)^2 \][/tex]
Here’s the step-by-step solution:
1. Expand the Expression:
[tex]\[ (\sin(\theta) - \cos(\theta))^2 \][/tex]
We use the algebraic identity for the square of a binomial: [tex]\((a - b)^2 = a^2 - 2ab + b^2\)[/tex]. In this case, [tex]\(a = \sin(\theta)\)[/tex] and [tex]\(b = \cos(\theta)\)[/tex].
2. Apply the Identity:
[tex]\[ (\sin(\theta) - \cos(\theta))^2 = \sin(\theta)^2 - 2\sin(\theta)\cos(\theta) + \cos(\theta)^2 \][/tex]
3. Combine Like Terms:
[tex]\[ \sin(\theta)^2 - 2\sin(\theta)\cos(\theta) + \cos(\theta)^2 \][/tex]
So, the simplified expression is:
[tex]\[ \sin(\theta)^2 - 2\sin(\theta)\cos(\theta) + \cos(\theta)^2 \][/tex]
