Question #5c2fd

1 Answer
Dec 23, 2017

#csc^2(a)-2csc(a)cot(a)+cot^2(a)#
or
#2cot^2(a)-2csc(a)cot(a)+1#
or
#2csc^2(a)-2csc(a)cot(a)-1#

Explanation:

We expand this just like a binomial, #(a-b)^2#:
#(csc(a)-cot(a))^2=csc^2(a)-2csc(a)cot(a)+cot^2(a)#

Now we have a couple of options because of the identity #1+cot^2(x)=csc^2(x)#.

Replacing #csc^2(a)#:

#csc^2(a)-2csc(a)cot(a)+cot^2(a) =#
# 1+cot^2(a)-2csc(a)cot(a)+cot^2(a) =#
#2cot^2(a)-2csc(a)cot(a)+1#

or go back and replace #cot^2(a)# with #csc^2(a)-1#:
#csc^2(a)-2csc(a)cot(a)+cot^2(a) =#
#csc^2(a)-2csc(a)cot(a)+csc^2(a)-1 =#
#2csc^2(a)-2csc(a)cot(a)-1#