Question

(csc x - 1) (csc x + 1)

Answer 1

Step-by-step explanation:

Here we have the following expression:

`(csc x - 1) (csc x + 1)`

So we need to extend this. Remember that the difference of squares tells us:

`(a-b)(a+b)=a^2-b^2`

So here:

`a=cscx \\ \\ b=1`

Thus:

`(csc x - 1) (csc x + 1)=csc^2x-1 \\ \\ \\ But: \\ \\ cscx=(1)/(sinx) \\ \\ \\ Then: \\ \\ csc^2x-1=(1)/(sin^2x)-1=(1-sin^2x)/(sin^2x)=(cos^2x)/(sin^2x)=cot^2x \\ \\ \\ Finally: \\ \\ \boxed{(csc x - 1) (csc x + 1)=cot^2x}`