To verify and simplify the given trigonometric equation:
[tex]\[ \sin \theta \cdot \cot \theta + \sin \theta \cdot \operatorname{cosec} \theta = 1 + \cos \theta \][/tex]
Let's break it down step by step.
1. Express cotangent and cosecant in terms of sine and cosine:
- [tex]\( \cot \theta = \frac{\cos \theta}{\sin \theta} \)[/tex]
- [tex]\( \operatorname{cosec} \theta = \frac{1}{\sin \theta} \)[/tex]
2. Substitute these identities into the left-hand side (LHS) of the equation:
[tex]\[
\sin \theta \cdot \cot \theta + \sin \theta \cdot \operatorname{cosec} \theta
= \sin \theta \cdot \frac{\cos \theta}{\sin \theta} + \sin \theta \cdot \frac{1}{\sin \theta}
\][/tex]
3. Simplify the expressions:
- The first term in the LHS simplifies as follows:
[tex]\[
\sin \theta \cdot \frac{\cos \theta}{\sin \theta} = \cos \theta
\][/tex]
- The second term in the LHS simplifies as follows:
[tex]\[
\sin \theta \cdot \frac{1}{\sin \theta} = 1
\][/tex]
4. Combine the simplified terms:
[tex]\[
\cos \theta + 1
\][/tex]
Now, the simplified left-hand side (LHS) is:
[tex]\[
\cos \theta + 1
\][/tex]
We see that both the simplified LHS and the right-hand side (RHS) are:
[tex]\[
1 + \cos \theta
\][/tex]
Therefore:
[tex]\[
\cos \theta + 1 = 1 + \cos \theta
\][/tex]
Thus, the equality holds true, verifying the given trigonometric equation:
[tex]\[
\sin \theta \cdot \cot \theta + \sin \theta \cdot \operatorname{cosec} \theta = 1 + \cos \theta
\][/tex]
