Question

csc x cot x (1-cos^2 x) = ______ x

@terenzreignz :)

Answer 1

idk this one :(

no answer choices! :/

Answer 2

Pythagorean identity is key here, along with a few more tricks...
\[\Large \cos^2(x) + \sin^2(x) = 1\]

Answer 3

okay :) how do i use that to solve? :/

Answer 4

Magic.

Answer 5

lol :P i love magic! :P

Answer 6

If we bring \(\large \cos^2(x)\) to the other side, we get
\[\Large \color{blue}{\sin^2(x) }= \color{red}{1-\cos^2(x)}\]

Answer 7

okay :) i see what you did there!

Answer 8

In that respect, this
\[\Large \csc(x)\cot(x)(\color{red}{1-\cos^2(x)})\]
becomes...?

Answer 9

csc(x)cot(x)(sin^2(x)) ?

Answer 10

yes, that's right. Now, let's break down the fancy csc and cot functions into simpler forms, and also, let's separate sin^2(x) into sin(x)sin(x)
\[\Large \frac{1}{\sin(x)}\cdot\frac{\cos(x)}{\sin(x)}\cdot\sin(x)\cdot\sin(x)\]

Answer 11

okay :)

Answer 12

Now... I'll just take my sword and do some slashing...
\[\Large \frac{1}{\cancel{\sin(x)}}\cdot\frac{\cos(x)}{{\sin(x)}}\cdot\cancel{\sin(x)}\cdot{\sin(x)}\]

Answer 13

And again...\[\Large \frac{1}{\cancel{\sin(x)}}\cdot\frac{\cos(x)}{\cancel{\sin(x)}}\cdot\cancel{\sin(x)}\cdot\cancel{\sin(x)}\]

Answer 14

hahaha thee sword is thy weapon :P

Answer 15

I use a Chinese Jian.

And now, we are left with
\[\huge \cos(x)\]

Answer 16

lol no clue what that is, but ima guess its some kind of super cool sword lol :P

okay! so the answer is just "cos" right? "cos" goes into the blank yeah? :)

Answer 17

okay haha googled it.. so its a sword that looks like it needs to be sheathed hahah :P cooool :)

Answer 18

Well, you have your answer :P

Answer 19

awesome!! :) yay! thankS!!! :D