Certainly! Let's delve into the given trigonometric identity step-by-step and verify its validity.
Problem Statement:
Verify the identity: [tex]\(\cos \theta \cdot \operatorname{cosec} \theta = \cot \theta\)[/tex].
### Step-by-Step Solution:
1. Trigonometric Functions Recap:
- [tex]\(\cos \theta\)[/tex] is the cosine of [tex]\(\theta\)[/tex].
- [tex]\(\operatorname{cosec} \theta\)[/tex] (or [tex]\(\csc \theta\)[/tex]) is the cosecant of [tex]\(\theta\)[/tex] which is the reciprocal of [tex]\(\sin \theta\)[/tex]. Hence, [tex]\(\csc \theta = \frac{1}{\sin \theta}\)[/tex].
- [tex]\(\cot \theta\)[/tex] is the cotangent of [tex]\(\theta\)[/tex] which is the ratio of [tex]\(\cos \theta\)[/tex] to [tex]\(\sin \theta\)[/tex]. Hence, [tex]\(\cot \theta = \frac{\cos \theta}{\sin \theta}\)[/tex].
2. Substitute the Definitions:
The left-hand side of the identity is:
[tex]\[
\cos \theta \cdot \csc \theta
\][/tex]
Substituting [tex]\(\csc \theta = \frac{1}{\sin \theta}\)[/tex]:
[tex]\[
\cos \theta \cdot \csc \theta = \cos \theta \cdot \frac{1}{\sin \theta}
\][/tex]
3. Simplify the Expression:
Simplify the product on the left-hand side:
[tex]\[
\cos \theta \cdot \frac{1}{\sin \theta} = \frac{\cos \theta}{\sin \theta}
\][/tex]
By definition, this is exactly the cotangent function:
[tex]\[
\frac{\cos \theta}{\sin \theta} = \cot \theta
\][/tex]
### Conclusion:
We've transformed the left-hand side of the given identity into the right-hand side through the use of fundamental trigonometric definitions. Thus, the identity is verified:
[tex]\[
\cos \theta \cdot \operatorname{cosec} \theta = \cot \theta
\][/tex]
In summary, the detailed steps confirm that the given trigonometric identity is indeed valid.
