Expert: Steve Holleran - 4/1/2007

Question
I'm falling behind in Precalc a bit, and I need some guidance on the following problems:

a.)
Simplify:
((cscx - 1)(cscx + 1)) / cos^2 x

b.)
Write in factored form:
4tan^2 x - (4 / cotx) + sinx cscx

c.)
Prove:
(cosx)(tanx + sinx cotx) = sinx + cos^2 x

d.)
Prove:
sin^2 x cos^3 x = (sin^2 x - sin^4 x)cosx

Please include the steps so I can actually learn how to do this. :)
Thanks!

Answer
Hi Andrew,

Okay, let's see what we've got here.

1.  If you multiply the numerator out, you get:

    csc^2 x - 1 / cos^2 x

=      cot^2 x / cos^2 x  [using one of the pythagorean
                                 identities]

= cos^2 x / sin^2 x *  1  /  cos^2 x  = 1/sin^2 x = csc^2 x


2.  If you write 4/ cot x in the middle as 4 tan x, you have :

 4 tan^2 x - 4 tan x + sin x * 1/sin x

= 4 tan^2 x - 4 tan x + 1 = (2 tan x - 1)^2


3.  Change tan x and cot x to sine and cosines:

cos x ( sin x / cos x + sin x * cos x / sin x)

= cos x ( sin x / cos x + cos x)  now multiply out:

= sin x + cos^2 x

4.  Break off one of the cosine powers:

sin^2 x * cos^3 x = sin^2 x * cos^2 x * cos x

Now use the identity for cos^2 x:

                 = sin^2 x * (1-sin^2 x) * cos x

                 = (sin^2 x - sin^4 x) * cos x


I hope this helps you out.

Steve Holleran