Question

How do you perform multiplication and use the fundamental identities to simplify `(cotx+cscx)(cotx-cscx)`?

Answer 1

`-1`

Explanation:

we start with the difference of squares expression

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

Applying that here we have

`(cotx+cscx)(cotx-cscx)=cot^2x-csc^2x`

now the identity connecting `cotx" & "cscx" is as follows"`

`cot^2x+1=csc^2x`

substituting for `csc^2x`

`=cot^2x-(cot^2x+1)`

`=cancel(cot^2x-cot^2x)-1`

`=-1`

Answer 2

`-1`

Explanation:

`"using the "color(blue)"trigonometric identities"`

`•color(white)(x)cotx=cosx/sinx" and "cscx=1/sinx`

`•color(white)(x)cos^2x=1-sin^2x`

`"expand the factors using the FOIL method"`

`rArr(cotx+cscx)(cotx-cscx)`

`=cot^2x-csc^2x`

`=cos^2x/sin^2x-1/sin^2xlarr" common denominator"`

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

`=(cancel(1)-sin^2xcancel(-1))/sin^2x`

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

`=-1`