How do you simplify $(1+costheta)(csctheta-cottheta)$ ?

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Answer 1 · 2017-01-17T20:27:42

When at a loss for what to do, it's a good idea to try writing everything using just $sintheta$ and $costheta$

$(1+costheta)(csctheta-cottheta) = (1+costheta)(1/sintheta-costheta/sintheta)$

We have a common denominator in the second factor, so let's write it as a single quotient.

$= (1+costheta)((1-costheta))/sintheta$

Now watch and pay attention. This is a good trick to know.

$= ((1+costheta)(1-costheta))/sintheta$

$= (1-cos^2theta)/sintheta$

And we can simplify some more,

$= (sin^2theta)/sintheta$

$= sin theta$

(for $sintheta != 0$ which is the same as: for $theta != pi k$ with $k$ an integer)

How do you simplify (1+costheta)(csctheta-cottheta)(1+costheta)(csctheta-cottheta)?