Let's solve the given expression step-by-step.
The given expression is:
[tex]\[
\frac{\sin^2(-\theta) - \cos^2(-\theta)}{\sin(-\theta) - \cos(-\theta)}
\][/tex]
We know the following trigonometric identities:
1. [tex]\(\sin(-\theta) = -\sin(\theta)\)[/tex]
2. [tex]\(\cos(-\theta) = \cos(\theta)\)[/tex]
Using these identities, we can rewrite the expression:
1. Substituting [tex]\(\sin(-\theta) = -\sin(\theta)\)[/tex] and [tex]\(\cos(-\theta) = \cos(\theta)\)[/tex]:
[tex]\[
\frac{\sin^2(-\theta) - \cos^2(-\theta)}{\sin(-\theta) - \cos(-\theta)} = \frac{(-\sin(\theta))^2 - \cos(\theta)^2}{-\sin(\theta) - \cos(\theta)}
\][/tex]
2. Simplify the squares in the numerator:
[tex]\[
= \frac{\sin^2(\theta) - \cos^2(\theta)}{-\sin(\theta) - \cos(\theta)}
\][/tex]
The simplified expression thus becomes:
[tex]\[
\text{Numerator} = \sin^2(\theta) - \cos^2(\theta)
\][/tex]
[tex]\[
\text{Denominator} = -\sin(\theta) - \cos(\theta)
\][/tex]
Considering the given simplified form:
[tex]\[
\cos \theta - \sin \theta
\][/tex]
First, let's compare the numerator:
[tex]\[
\sin^2(\theta) - \cos^2(\theta)
\][/tex]
To analyze further, notice:
[tex]\[
\sin^2(\theta) - \cos^2(\theta) = -(\cos^2(\theta) - \sin^2(\theta))
\][/tex]
So using trigonometric identity if rewritten:
[tex]\[
= -\cos(2\theta)
\][/tex]
Next, the denominator:
[tex]\[
- (\sin(\theta) + \cos(\theta)) = - (\sin(\theta) + \cos(\theta))
\][/tex]
Now express the entire fraction:
[tex]\[
\frac{-\cos (2\theta)}{- (\sin(\theta) + \cos(\theta))}
\][/tex]
If simplified further, the numerator has cos(\theta + π/4) with:
Combining:
The simplified step, identify denominator:
After all steps, given expression comparing:
Thus we conclude the given result does not match directly. So:
\[
(\cos(\theta) - \sin(\theta ))\ is simplified vs \sqrt{2} (cos......)
Therefore, the comparison ultimately shows expressions unequal directly.
Conclusively confirmed with result, the expressions evaluated distinctly non-equal.
