To solve the given equation, let's start by rewriting and simplifying the expression:
[tex]\[
\sin \theta \cos ^3 \theta + \sin ^3 \theta \cos \theta = \sin \theta \cos \theta
\][/tex]
First, notice that [tex]\(\sin \theta \cos \theta\)[/tex] appears on both sides of the equation. We can subtract [tex]\(\sin \theta \cos \theta\)[/tex] from both sides to simplify:
[tex]\[
\sin \theta \cos ^3 \theta + \sin ^3 \theta \cos \theta - \sin \theta \cos \theta = 0
\][/tex]
Now, factor out the common term [tex]\(\sin \theta \cos \theta\)[/tex] from the left-hand side:
[tex]\[
\sin \theta \cos \theta (\cos ^2 \theta + \sin ^2 \theta - 1) = 0
\][/tex]
Recall the Pythagorean identity which states:
[tex]\[
\sin^2 \theta + \cos^2 \theta = 1
\][/tex]
Substitute [tex]\(1\)[/tex] for [tex]\(\sin^2 \theta + \cos^2 \theta\)[/tex] in the equation:
[tex]\[
\sin \theta \cos \theta (1 - 1) = 0
\][/tex]
This simplifies to:
[tex]\[
\sin \theta \cos \theta \cdot 0 = 0
\][/tex]
Thus, no matter what values [tex]\(\theta\)[/tex] takes, the left-hand side will always be zero:
[tex]\[
0 = 0
\][/tex]
Therefore, the simplified form of the original equation is indeed:
[tex]\[
0
\][/tex]
This completes the verification and confirmation that the original expression simplifies to zero.
