Expert: Sherman D. - 12/31/2006

Question
Please help me!!

a) simplify completely:
tanx + cotx / csc^2 x

b)simplify completely:
sin(pi + x) -sin(pi-x)

I worked both out but did I do it right?

a) sinx/cosx + cosx/sinx divided by (1/sinx)^2
  sin^2 x + cos^2 x / sinx cosx divided by (1/sin^2 x)
  1/sinx cosx multiplied by sin^2 x / 1
  sin^2 x / sin x cos x
  cancel sin^2 x on top with sin x on bottom
  sin x / cos x
  ****FINAL ANSWER: tan x****
Is this correct?

b) (sin pi cos x + cos pi sin x) - (sin pi cos x - cos pi sin x)
  sin pi cos x + cos pi sin x
- sin pi cos x - cos pi sin x
--------------------------------
   0             +2cos pi
******Final ANSWER: 2cos pi *******
Is this right? I am afraid that I may have done too much canceling?

Please help me! Thanks so much.


Answer
"a" is correct, but "b" isn't quite correct.

just change the signs of the +cos and -cos, and change cos(x) to sin(x) and you get -sin(x) - sin(x).

Keep in mind cos(pi) = -1

so to write 2cos(pi) would give you -2

But don't feel bad, i had the answer at first as -2cos(x) instead of -2sin(x), i had the correct sign, but wrong trig.

Here is my step by step.

a)
(tan(x) + cot(x))/(csc(x)^2)
((sin(x)/cos(x)) + (cos(x)/sin(x))) / (1/sin(x)^2)
((sin(x)^2 + cos(x)^2)/(sin(x)cos(x))) / (1/sin(x)^2)
(1/(sin(x)cos(x))) / (1/sin(x)^2)
(1/(sin(x)cos(x))) * sin(x)^2
(sin(x)^2)/(sin(x)cos(x))
(sin(x))/(cos(x))
tan(x)

(tan(x) + cot(x))/(csc(x)^2) = tan(x)

-----------------------------------------------------------

b)
sin(pi + x) - sin(pi - x)

sin(pi + x) = sin(pi)cos(x) + sin(x)cos(pi) = -sin(x)
sin(pi - x) = sin(pi)cos(x) - sin(x)cos(pi) = sin(x)

sin(pi + x) - sin(pi - x) = -cos(x) - cos(x) = -2sin(x)

-----------------------------------------------------------

Info found at

www.math.com/tables/trig/identities.htm

i also used www.calculator.com/calcs/GCalc.html

to check my answer. First i went one way, then i went another way, until i got the answer i was suppose to get.