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

2 Answers
sjc
Aug 22, 2017

-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

Jim G.
Aug 22, 2017

-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

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