Sure, let's solve the given trigonometric expression step by step:
Given expression:
[tex]\[
\frac{\operatorname{cosec} \theta}{\sin \theta} - \frac{\cot \theta}{\tan \theta} = 1
\][/tex]
First, recall the trigonometric identities:
[tex]\[
\operatorname{cosec} \theta = \frac{1}{\sin \theta}
\][/tex]
[tex]\[
\cot \theta = \frac{\cos \theta}{\sin \theta}
\][/tex]
[tex]\[
\tan \theta = \frac{\sin \theta}{\cos \theta}
\][/tex]
Now, substitute these identities into the expression:
For [tex]\(\frac{\operatorname{cosec} \theta}{\sin \theta}\)[/tex]:
[tex]\[
\frac{\operatorname{cosec} \theta}{\sin \theta} = \frac{\frac{1}{\sin \theta}}{\sin \theta} = \frac{1}{\sin \theta \cdot \sin \theta} = \frac{1}{\sin^2 \theta}
\][/tex]
For [tex]\(\frac{\cot \theta}{\tan \theta}\)[/tex]:
[tex]\[
\frac{\cot \theta}{\tan \theta} = \frac{\frac{\cos \theta}{\sin \theta}}{\frac{\sin \theta}{\cos \theta}} = \frac{\cos \theta}{\sin \theta} \cdot \frac{\cos \theta}{\sin \theta} = \frac{\cos^2 \theta}{\sin^2 \theta}
\][/tex]
So the original expression now becomes:
[tex]\[
\frac{1}{\sin^2 \theta} - \frac{\cos^2 \theta}{\sin^2 \theta}
\][/tex]
Combine the terms over a common denominator:
[tex]\[
\frac{1 - \cos^2 \theta}{\sin^2 \theta}
\][/tex]
Recall the Pythagorean identity:
[tex]\[
1 - \cos^2 \theta = \sin^2 \theta
\][/tex]
Substitute this identity back into the expression:
[tex]\[
\frac{\sin^2 \theta}{\sin^2 \theta} = 1
\][/tex]
Thus, we have shown step-by-step that the given expression simplifies to:
[tex]\[
1
\][/tex]
Hence, it is true that:
[tex]\[
\frac{\operatorname{cosec} \theta}{\sin \theta} - \frac{\cot \theta}{\tan \theta} = 1
\][/tex]
