To solve the given trigonometric identity [tex]\(\sec \theta \cdot \cot \theta = \operatorname{cosec} \theta\)[/tex], we'll break it down step-by-step using known trigonometric identities:
1. Rewrite the trigonometric functions using basic trigonometric identities:
- We know that:
[tex]\[
\sec \theta = \frac{1}{\cos \theta}
\][/tex]
[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 of the given equation:
Substitute [tex]\(\sec \theta\)[/tex] and [tex]\(\cot \theta\)[/tex] into the product:
[tex]\[
\sec \theta \cdot \cot \theta = \left( \frac{1}{\cos \theta} \right) \left( \frac{\cos \theta}{\sin \theta} \right)
\][/tex]
3. Simplify the expression:
Simplify the product step-by-step:
[tex]\[
\left( \frac{1}{\cos \theta} \right) \cdot \left( \frac{\cos \theta}{\sin \theta} \right) = \frac{1}{\cos \theta} \times \frac{\cos \theta}{\sin \theta}
\][/tex]
Notice that [tex]\(\cos \theta\)[/tex] in the numerator and denominator cancel each other out:
[tex]\[
\frac{\cos \theta}{\cos \theta} = 1
\][/tex]
Hence, we have:
[tex]\[
\frac{1}{\sin \theta}
\][/tex]
4. Relate the result to the right-hand side:
We know that:
[tex]\[
\frac{1}{\sin \theta} = \operatorname{cosec} \theta
\][/tex]
5. Conclusion:
Therefore,
[tex]\[
\sec \theta \cdot \cot \theta = \operatorname{cosec} \theta
\][/tex]
Thus, the trigonometric identity [tex]\(\sec \theta \cdot \cot \theta = \operatorname{cosec} \theta\)[/tex] is true.
