How do you simplify # (1-cos^2 theta)(1+cot^2 theta) #?

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Answer 1 · 2016-06-01T05:13:12

It simplifies to 1

We'll use a couple of trig identities to solve this:

#sin^2theta+cos^2theta=1#
#1+cot^2theta=csc^2theta#

We can substitute #csc^2theta# directly into the second term but need to do a bit of math to get the substitution for the first term:

#sin^2theta+cos^2theta=1#

#sin^2theta=1-cos^2theta#

Now let's substitute:

#(1-cos^2theta)(1+cot^2theta)#

#(sin^2theta)(csc^2theta)#

The relationship between sin and csc is:

#csc = 1/sin#

thus

#csc^2theta=1/sin^2theta#

and so #(sin^2theta)(1/sin^2theta)=1#