How do you simplify $costheta/(sin^2theta-1)$ ?

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Answer 1 · 2016-11-15T04:29:53

$-sectheta$ , $theta!=pi/2+-n2pi$ where n is a natural number

We have:

$costheta/(sin^2theta-1)$

Remember that there is an identity:

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

and so

$cos^2theta=-sin^2theta+1$ which can be written as:

$cos^2theta=-(sin^2theta-1)$ and so

$-cos^2theta=(sin^2theta-1)$

so let's substitute that into our original statement:

$costheta/(sin^2theta-1)$

$costheta/(-cos^2theta)$

and now we can simplify:

$1/(-costheta)=-sectheta$

Ok, so now what are the limitations? It's $sin^2theta!=1$ (that would make the denominator in our original question be 0) and so:

$theta!=pi/2+-n2pi$ where n is a natural number

How do you simplify costheta/(sin^2theta-1)costheta/(sin^2theta-1)?