Question

Prove the identity note that each statement must be based

Answer 1

Given the identity:

`(\csc ^2x-1)\sin x=\cos ^2x\csc x`

We will prove the identity as follows

Starting from the left-hand side

First, use the rule of reciprocal

`(\csc ^2x-1)\sin x=((1)/(\sin^2x)-1)\sin x`

Second, use the rule of quotient

`=((1-\sin^2x)/(\sin^2x))\sin x`

Third, use the Pythagorean theorem

`=(\cos^2x)/(\sin^2x)\cdot\sin x`

Fourth, Multiply using the rule algebra

`=(cos^2x)/(\sin x)`

Finally, use the rule of reciprocal

`=\cos ^2x\csc x`

Which will be matched with the right-hand side.