How do you simplify cot^2x-csc^2xcot2xcsc2x?

3 Answers
Jan 3, 2018

-11

Explanation:

One of the fundamental identities is 1+cot^2(x) = csc^2(x)1+cot2(x)=csc2(x).

Starting with your given:

cot^2(x)-csc^2(x)cot2(x)csc2(x)

Replace csc^2(x)csc2(x) with 1+cot^2(x)1+cot2(x):

cot^2(x)-(1+cot^2(x))cot2(x)(1+cot2(x))

=cot^2(x)-1-cot^2(x))=cot2(x)1cot2(x))

=-1=1

Jan 3, 2018

cot^2x-csc^2x=-1cot2xcsc2x=1

Explanation:

Use the identity 1+cot^2x=csc^2x1+cot2x=csc2x

Subtract cot^2xcot2x to both sides:

1=csc^2x-cot^2x1=csc2xcot2x

Rewrite as:

1=-cot^2x+csc^2x1=cot2x+csc2x

Divde both sides by a negative (-)()

-1=cot^2x-csc^2x1=cot2xcsc2x

Jan 3, 2018

-11

Explanation:

Another way is to reduce all the functions to sine and cosines

cot^2x-csc^2x=cos^2x/sin^2x-1/sin^2xcot2xcsc2x=cos2xsin2x1sin2x

=(cos^2x-1)/sin^2x=cos2x1sin2x

= -(1-cos^2x)/sin^2x=-sin^2x/sin^2x=1cos2xsin2x=sin2xsin2x

=-1=1