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 $(sin^2theta)(1/sin^2theta)=1$

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