Sure, let's walk through the steps to verify the given trigonometric identity step-by-step, proving that \(\frac{\operatorname{cosec} \theta}{\tan \theta + \cot \theta} = \cos \theta\).
1. Start with the left-hand side (LHS):
[tex]\[
\frac{\operatorname{cosec} \theta}{\tan \theta + \cot \theta}
\][/tex]
2. Recall the trigonometric identities:
- \(\operatorname{cosec} \theta = \frac{1}{\sin \theta}\)
- \(\tan \theta = \frac{\sin \theta}{\cos \theta}\)
- \(\cot \theta = \frac{\cos \theta}{\sin \theta}\)
3. Substitute these identities into the LHS:
[tex]\[
\frac{\frac{1}{\sin \theta}}{\frac{\sin \theta}{\cos \theta} + \frac{\cos \theta}{\sin \theta}}
\][/tex]
4. Simplify the denominator:
[tex]\[
\frac{\sin \theta}{\cos \theta} + \frac{\cos \theta}{\sin \theta}
= \frac{\sin^2 \theta + \cos^2 \theta}{\sin \theta \cos \theta}
\][/tex]
5. Recall the Pythagorean identity, which states:
[tex]\[
\sin^2 \theta + \cos^2 \theta = 1
\][/tex]
So the denominator becomes:
[tex]\[
\frac{1}{\sin \theta \cos \theta}
\][/tex]
6. Rewrite the original fraction with the simplified denominator:
[tex]\[
\frac{\frac{1}{\sin \theta}}{\frac{1}{\sin \theta \cos \theta}}
\][/tex]
7. Simplify this expression by multiplying numerator and denominator by \(\sin \theta \cos \theta\):
[tex]\[
= \frac{1}{\cos \theta}
\][/tex]
8. The resulting expression is:
[tex]\[
\cos \theta
\][/tex]
So the left-hand side simplifies directly to \(\cos \theta\), verifying that
[tex]\[
\frac{\operatorname{cosec} \theta}{\tan \theta + \cot \theta} = \cos \theta
\][/tex]
Hence, the trigonometric identity is proven to be true.
