Question

Complete the identity

Answer 1

Explanation:

Given expression is
`(\left(\csc\left(x\right)+1\right)\left(\csc\left(x\right)-1\right))/(\cot^2\left(x\right))`

Now we need to simplify that to complete the identity.

`(\left(\csc\left(x\right)+1\right)\left(\csc\left(x\right)-1\right))/(\cot^2\left(x\right))`

`=(\csc^2\left(x\right)+1\csc\left(x\right)-1\csc\left(x\right)-1^2)/(\cot^2\left(x\right))`

`=(\csc^2\left(x\right)+1\csc\left(x\right)-1\csc\left(x\right)-1)/(\cot^2\left(x\right))`

`=(\csc^2\left(x\right)-1)/(\cot^2\left(x\right))`

Apply formula

`\csc^2\left(\theta\right)=1+\cot^2\left(\theta\right)`

`=(\cot^2\left(x\right))/(\cot^2\left(x\right))`

`=1`

Hence required identity is

`(\left(\csc\left(x\right)+1\right)\left(\csc\left(x\right)-1\right))/(\cot^2\left(x\right))=1`

Answer 2

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

EXPLANATION

The given identity is:

`(( \csc x + 1)( \csc(x) - 1))/( \cot ^(2) (x) )`

Recall that:

`(x + 1)(x - 1) = {x}^(2) - {y}^(2)`

We apply difference of two squares to the numerator to get:

`(\csc ^(2) x - 1)/( \cot ^(2) (x) )`

Also recall the Pythagorean Identity.

`1 + \cot^(2) (x) = \csc ^(2) (x)`

This implies that,.

`\cot^(2) (x) = \csc ^(2) (x) - 1`

Hence our identity becomes:

`(\cot ^(2) x )/( \cot ^(2) (x) ) = 1`