To solve the equation [tex]\( \frac{\cot^2 t}{\csc t} = \csc t - \sin t \)[/tex], let's break it down step by step, considering trigonometric identities and simplifications.
First, recall the definitions of the trigonometric functions involved:
[tex]\[
\cot t = \frac{\cos t}{\sin t}
\][/tex]
[tex]\[
\csc t = \frac{1}{\sin t}
\][/tex]
Given the left-hand side (LHS) of the equation:
[tex]\[
\frac{\cot^2 t}{\csc t}
\][/tex]
Substitute the definitions of [tex]\(\cot t\)[/tex] and [tex]\(\csc t\)[/tex]:
[tex]\[
\cot^2 t = \left( \frac{\cos t}{\sin t} \right)^2 = \frac{\cos^2 t}{\sin^2 t}
\][/tex]
[tex]\[
\csc t = \frac{1}{\sin t}
\][/tex]
Now substitute these into the left-hand side:
[tex]\[
\frac{\cot^2 t}{\csc t} = \frac{\frac{\cos^2 t}{\sin^2 t}}{\frac{1}{\sin t}} = \frac{\cos^2 t}{\sin^2 t} \cdot \sin t = \frac{\cos^2 t \cdot \sin t}{\sin^2 t} = \frac{\cos^2 t}{\sin t}
\][/tex]
So the simplified form of the left-hand side is:
[tex]\[
\frac{\cos^2 t}{\sin t}
\][/tex]
Now consider the right-hand side (RHS) of the equation:
[tex]\[
\csc t - \sin t
\][/tex]
Using the definition of [tex]\(\csc t\)[/tex]:
[tex]\[
\csc t = \frac{1}{\sin t}
\][/tex]
Thus, the right-hand side can be expressed as:
[tex]\[
\frac{1}{\sin t} - \sin t
\][/tex]
So, we have the simplified forms:
[tex]\[
\text{LHS} = \frac{\cos^2 t}{\sin t}
\][/tex]
[tex]\[
\text{RHS} = \frac{1}{\sin t} - \sin t
\][/tex]
Now we can write the given equation in its simplified form:
[tex]\[
\frac{\cos^2 t}{\sin t} = \frac{1}{\sin t} - \sin t
\][/tex]
Therefore, the simplified forms of both sides of the equation are:
[tex]\[
\boxed{\left( \frac{\cos^2 t}{\sin t}, \frac{1}{\sin t} - \sin t \right)}
\][/tex]
