Given the identity:
[tex](\csc ^2x-1)\sin x=\cos ^2x\csc x[/tex]
We will prove the identity as follows
Starting from the left-hand side
First, use the rule of reciprocal
[tex](\csc ^2x-1)\sin x=(\frac{1}{\sin^2x}-1)\sin x[/tex]
Second, use the rule of quotient
[tex]=(\frac{1-\sin^2x}{\sin^2x})\sin x[/tex]
Third, use the Pythagorean theorem
[tex]=\frac{\cos^2x}{\sin^2x}\cdot\sin x[/tex]
Fourth, Multiply using the rule algebra
[tex]=\frac{cos^2x}{\sin x}[/tex]
Finally, use the rule of reciprocal
[tex]=\cos ^2x\csc x[/tex]
Which will be matched with the right-hand side.
