Expert: Jack Cheng - 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             +2B2cos 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
Hi Jamie,

The first one is absolutely correct :)

The second one is a little bit off. When you subtracted, you cannot just get rid of sin x because it's multiplied. It should have been 2 cos pi sin x, and since cos pi is -1, the final answer should be -2 sin x (assuming radian measures, if in degrees, it's different).

For the second one, you could also have solved it geometrically. sin(pi+x) and sin(pi-x) are the same values, with the exception of the sign. sin(pi+x) is the same as the -sin(x) because the opposite side is negative. sin(pi-x) is the same as sin(x). So sin(pi+x)-sin(pi-x) = -sin(x)-sin(x) = -2sin(x).

You can always check your work by plugging some random value for x (zero is usually my favorite) and see if what you got equals the original problem. For the first one, I used 1 for the x (zero won't work as cot(0) is infinite) to check that your work is correct.

I hope this helps,
Jack