Sure, let's verify the given trigonometric identity step-by-step.
Given equation:
[tex]\[
\sin \theta \cdot \cot \theta + \sin \theta \cdot \csc \theta = 1 + \cos \theta
\][/tex]
First, let's rewrite [tex]\(\cot \theta\)[/tex] and [tex]\(\csc \theta\)[/tex] in terms of [tex]\(\sin \theta\)[/tex] and [tex]\(\cos \theta\)[/tex]:
[tex]\[
\cot \theta = \frac{\cos \theta}{\sin \theta}
\][/tex]
[tex]\[
\csc \theta = \frac{1}{\sin \theta}
\][/tex]
Substitute these into the given equation:
[tex]\[
\sin \theta \cdot \frac{\cos \theta}{\sin \theta} + \sin \theta \cdot \frac{1}{\sin \theta}
\][/tex]
Now, simplify each term separately:
[tex]\[
\frac{\sin \theta \cdot \cos \theta}{\sin \theta} + \frac{\sin \theta \cdot 1}{\sin \theta}
\][/tex]
Since [tex]\(\sin \theta \neq 0\)[/tex], we can cancel [tex]\(\sin \theta\)[/tex] in both terms:
[tex]\[
\cos \theta + 1
\][/tex]
Thus, the left side of the equation simplifies to:
[tex]\[
\cos \theta + 1
\][/tex]
Compare this with the right side of the original equation:
[tex]\[
1 + \cos \theta
\][/tex]
We see that:
[tex]\[
\cos \theta + 1 = 1 + \cos \theta
\][/tex]
Thus, the given equation simplifies correctly and verifies the identity:
[tex]\[
\sin \theta \cdot \cot \theta + \sin \theta \cdot \csc \theta = 1 + \cos \theta
\][/tex]
So, the given trigonometric identity is correct and verified.
