Expert: Paul Klarreich - 10/15/2008

Question
I am having trouble vcerifying this identity because of the powers for each function.
sin^6(x)+cos^6(x)=sin^4(x)-sin^2(x)cos^2(x)+cos^4
The other one I was having a hard time with was
(csc(x)/cot(x))-(cot(x)/csc(x))=tan(x)sin(x).

Answer
Questioner:   Stephanie
Category:  Advanced Math
Private:  No
 
Subject:  Verifying Trig Functions
Question:  I am having trouble verifying this identity because of the powers for each function.

sin^6(x)+cos^6(x)=sin^4(x)-sin^2(x)cos^2(x)+cos^4

The other one I was having a hard time with was

(csc(x)/cot(x))-(cot(x)/csc(x))=tan(x)sin(x).
..................................
Hi, Stephanie,

BTW, 'verifying' is not the correct word here -- you are proving these identities.

These three (yes, three) use the Pythagorean Identities;

sin^2 + cos^2 = 1  (sorry about leaving out the x's)
sec^2 - tan^2 = 1
csc^2 - cot^2 = 1

and variations like bringing a term to the other side.
.........................

For the first one,  use this factoring identity for 'sum of cubes'.

a^3 + b^3 = (a + b)(a^2 - ab + b^2)

For your identity:

sin^6(x)+cos^6(x) = sin^4(x)-sin^2(x)cos^2(x)+cos^4(x)

[I will abbreviate:  s = sin x,  c = cos x, to save typing and make thing a little more transparent.]

Left side:

s^6 + c^6

(s^2 + c^2)(s^4 - s^2c^2 + c^4)

(     1   )(s^4 - s^2c^2 + c^4)

done.
....................
Your second one:

(csc(x)/cot(x))-(cot(x)/csc(x))=tan(x)sin(x).

WAIT A MINUTE!   That is not the one you sent earlier.  Pretty sneaky!

Anyway the one you sent earlier was:

sex(x)+tan(x)=1/(sec(x)-tan(x))

which goes like this:
                      1
sec(x)+tan(x)  = -----------------
                sec(x) - tan(x)

                      1         sec(x) + tan(x)
sec(x)+tan(x)  = --------------- ----------------
                sec(x) - tan(x) sec(x) + tan(x)

                 sec(x) + tan(x)
sec(x)+tan(x)  = -------------------
                sec^2(x) - tan^2(x)


                 sec(x) + tan(x)
sec(x)+tan(x)  = ----------------
                       1

Done.
.....................................

For this one:

(csc(x)/cot(x))-(cot(x)/csc(x))  = tan(x)sin(x).

Rewrite it:

csc x       cot x
------  -  ------- = tan x sin x
cot x       csc x

Suggestion:  Combine fractions on the left:

csc^2(x) - cot^2(x)
-------------------
 cot x csc x

        1
-------------------
 cot x csc x

        1
-------------------
(1/tan x) (1/sin x)

tan x sin x

Done.

Suggestion:  If you don't like working with tan, cot, sec, csc, start by changing all of them to sines and cosines.  There might be more algebra (with fractions) to do, but less to remember.