How do you simplify #csc^2(θ) / (1 + tan^2(θ) )#?
Let's see, we have:
In this case, I would probably start by rewriting everything in terms of sines and cosines, since we know that
So we now have:
Here, I notice that the denominator can be simplified if I would multiply it by
and if I multiply
Opening the parentheses, this becomes:
which is our famous identity and is
Yes. The denominator becomes 1 if I would multiply by
So let's do the following:
We can do this since any number over itself (except zero) is equal to 1.
So provided that
Now, we know the denominator becomes 1, and we are left with just the "top":
Use the trig identity: