SOLUTION: How do you simplify (1+sec(-x))/(sin(-x)+tan(-x)) without leaving a fraction?

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Question 716501: How do you simplify
(1+sec(-x))/(sin(-x)+tan(-x))
without leaving a fraction?
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
%281%2Bsec%28-x%29%29%2F%28sin%28-x%29%2Btan%28-x%29%29+
First let's use the even/odd properties of the Trig functions. f(-x) = f(x) for even functions and f(-x) = -f(x) for odd functions. cos and sec are even and the rest are odd. So:
%281%2Bsec%28x%29%29%2F%28-sin%28x%29%2B%28-tan%28x%29%29%29+
Factoring out -1 in the denominator:
%281%2Bsec%28x%29%29%2F%28-1%2A%28sin%28x%29%2Btan%28x%29%29%29+
which is equal to:
-%281%2Bsec%28x%29%29%2F%28sin%28x%29%2Btan%28x%29%29+

Now we'll try to simplify and eliminate the fraction. A good rule of thumb in Trig is: When you don't see anything better (and you really should look first), then change any sec's, csc's, tan's or cot's into expressions of sin's and/or cos's. I don't see anything else that is obvious so:
-%281%2B1%2Fcos%28x%29%29%2F%28sin%28x%29%2Bsin%28x%29%2Fcos%28x%29%29+

We can eliminate the fractions within a fraction by multiplying the numerator and denominator by cos(x):

which simplifies to:
-%28cos%28x%29%2B1%29%2F%28sin%28x%29%2Acos%28x%29%2Bsin%28x%29%29+
Factoring out sin(x) in the denominator:
-%28cos%28x%29%2B1%29%2F%28sin%28x%29%2A%28cos%28x%29%2B1%29%29+
The (cos(x)+1)'s cancel:
-1%2Fsin%28x%29
And finally, since 1/sin(x) = csc(x):
-csc%28x%29