establishing identities with ^3 and ^4 in them

[bikeman]

i am asked two similar problems. my question is how it was done as the example in the book i am reading is useless in explaining.
the first question is:
(sin^3t+cos^3t)/(sint+cost)=1-sint cost
the example plays out to what i assume is factoring sint+cost out of it to look like (sint+cost)(sin^2t-sint cost+cos^2)/(sint+cost)

my question is, where did the "sin^2t-sint cost+cos^2" come from? when i factor it it doesnt look like that.

the next problem is (cos^4t-sin^4t)/(cos[2t]) how do i get rid of the ^4?


thanks for any help provided.

[Martingale]

i am asked two similar problems. my question is how it was done as the example in the book i am reading is useless in explaining.
the first question is:
(sin^3t+cos^3t)/(sint+cost)=1-sint cost
the example plays out to what i assume is factoring sint+cost out of it to look like (sint+cost)(sin^2t-sint cost+cos^2)/(sint+cost)

my question is, where did the "sin^2t-sint cost+cos^2" come from? when i factor it it doesnt look like that.

the next problem is (cos^4t-sin^4t)/(cos[2t]) how do i get rid of the ^4?


thanks for any help provided.

[stapel_eliz]

where did the "sin^2t-sint cost+cos^2" come from? when i factor it it doesnt look like that.

How did you factor? What did you get?

(To review how to factor differences of squares and sums and differences of cubes, try here).

[bikeman]

I factored wrong.
i found that (cos^4t-sin^4t) is the same as saying (cos^2t+sin^2t)(cos^2t-sin^2t) one of which is a Pythagorean identity. the lower part of the problem, Cos(2t) is part of a series of identities which which cancels one half of the upper part of the equation. meaning you are left is 1 X1 which equals 1