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

Question 716501: How do you simplify
(1+sec(-x))/(sin(-x)+tan(-x))
without leaving a fraction?
Answer by jsmallt9(3758) (Show Source): You can put this solution on YOUR website!

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:

Factoring out -1 in the denominator:

which is equal to:

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:

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

which simplifies to:

Factoring out sin(x) in the denominator:

The (cos(x)+1)'s cancel:

And finally, since 1/sin(x) = csc(x):

RELATED QUESTIONS
(sec x + tan x)(1 - sin... (answered by jrfrunner)

tan(x) [sec(x) - cos(x)] ________________________ sin (x)... (answered by mananth)

How do you solve these using trigonometric identities? (This is a fraction by the way): (answered by MathLover1)

Simplify... (answered by Alan3354)

Simplify tan(x - π). Select one: a. tan(x) b. sec(x) c. sin(x) d. cos(x) (answered by stanbon)

Verify: cos(x) / 1+sin(x) = sec(x) - tan(x) (answered by stanbon)

(sin x-1)(tan x+sec x) = cos... (answered by fractalier)

(sec(x)-cos(x))(csc(x)-sin(x))=tan(x)/1+tan^2(x) please prove the identity,. do only one... (answered by Edwin McCravy)

Simplify: sin(x) /... (answered by jim_thompson5910)