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

Question

41325 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2018-01-03T00:16:47

#-1#

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

Starting with your given:

#cot^2(x)-csc^2(x)#

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

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

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

#=-1#

Answer 2 · 2018-01-03T00:27:40

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

Use the identity #1+cot^2x=csc^2x#

Subtract #cot^2x# to both sides:

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

Rewrite as:

#1=-cot^2x+csc^2x#

Divde both sides by a negative #(-)#

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

Answer 3 · 2018-01-03T09:51:16

#-1#

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

#cot^2x-csc^2x=cos^2x/sin^2x-1/sin^2x#

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

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

#=-1#