Let's examine and solve the given equation step-by-step.
The equation given is:
[tex]\[
\frac{\sin^3 \theta + \cos^3 \theta}{\sin \theta + \cos \theta} = \frac{1 - 1}{2} \sin(2\theta)
\][/tex]
### Step 1: Simplify the right-hand side
First, simplify the right-hand side of the equation:
[tex]\[
\frac{1 - 1}{2} \sin(2\theta)
\][/tex]
Since [tex]\(1 - 1\)[/tex] equals 0, the whole expression simplifies to:
[tex]\[
0
\][/tex]
Thus, the equation now is:
[tex]\[
\frac{\sin^3 \theta + \cos^3 \theta}{\sin \theta + \cos \theta} = 0
\][/tex]
### Step 2: Analyze the left-hand side
Next, let's simplify the left-hand side of the equation. The expression [tex]\(\frac{\sin^3 \theta + \cos^3 \theta}{\sin \theta + \cos \theta}\)[/tex] can be understood by rewriting the numerator.
Using the algebraic identity [tex]\(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\)[/tex], with [tex]\(a = \sin \theta\)[/tex] and [tex]\(b = \cos \theta\)[/tex], we have:
[tex]\[
\sin^3 \theta + \cos^3 \theta = (\sin \theta + \cos \theta)(\sin^2 \theta - \sin \theta \cos \theta + \cos^2 \theta)
\][/tex]
Now substitute this into the expression:
[tex]\[
\frac{(\sin \theta + \cos \theta)(\sin^2 \theta - \sin \theta \cos \theta + \cos^2 \theta)}{\sin \theta + \cos \theta}
\][/tex]
Since [tex]\(\sin \theta + \cos \theta\)[/tex] is not zero, we can cancel it from the numerator and the denominator:
[tex]\[
\sin^2 \theta - \sin \theta \cos \theta + \cos^2 \theta
\][/tex]
Using the Pythagorean identity, [tex]\(\sin^2 \theta + \cos^2 \theta = 1\)[/tex], we get:
[tex]\[
1 - \sin \theta \cos \theta
\][/tex]
However, we must verify if the left-hand side simplifies further:
Through further simplification or examination, we can conclude that:
[tex]\[
\sin^3 \theta + \cos^3 \theta = (\sin \theta + \cos \theta)(1 - \sin \theta \cos \theta)
\][/tex]
So, the simplified form of the left-hand side is:
[tex]\[
1 - \frac{\sin(2 \theta)}{2}
\][/tex]
However, this needs to be zero according to our original simplification, therefore this implies that:
[tex]\[
1 - \frac{\sin(2 \theta)}{2} = 0
\][/tex]
### Final Answer:
Hence:
[tex]\[
\frac{\sin ^3 \theta+\cos ^3 \theta}{\sin \theta+\cos \theta}= 0
\][/tex]
