Question

How do I prove the following function?
cot2^x/cscx+1 = 1-sinx/sinx

Answer 1

do you have your trig formula sheet handy?

Answer 2

My what?

Answer 3

If you mean my identities, yes.

Answer 4

trigonometry formulas cheatsheet :)

Answer 5

so you can check the trigonometric identities so you can factor them out

Answer 6

Yeah I have my identities with me.

Answer 7

ok, check the pythagorean identities, specifically the one for csc^2

Answer 8

and from there, get what cot^2 is :)

Answer 9

I can replace cot^2x with 1-csc^2x

Answer 10

mmm, csc^2-1 you meant

Answer 11

Oh, my bad.

Answer 12

csc^2x-1/cscx+1 <--- That's it lol

Answer 13

yes
now keep in mind that \[csc(x)^2-1= csc(x)^2-1^2\] which is also the same as \[(csc(x)-1)(csc(x)+1)\]

Answer 14

(cscx-1)(cscx+1) / cscx+1

So that would simplify to just cscx-1 on the left?

Answer 15

yes

Answer 16

Now there's 1-sinx/sinx

Answer 17

and that \[\frac{a-b}{c}=\frac{a}{c}-\frac{b}{c}\]

Answer 18

1/sin x - sin x/sin x ?

Answer 19

yes

Answer 20

So that will end up as just 1/sin x ?

Answer 21

well, sinx/sinx isn't "0"

Answer 22

1/sinx+1?

Answer 23

\[\frac{1}{sin(x)}+1\]

Answer 24

So that leaves us at:
cscx-1 = 1/sinx +1
?

Answer 25

yes, check your trig sheet for the "cofunction identities" :)

Answer 26

I have those.

Answer 27

there is your answer :)

Answer 28

They all deal with pi/2 though. How do I use them here?

Answer 29

ohh shoot, yes, my bad, I meant "reciprocal identities"

Answer 30

cscx = 1/sinx,

Answer 31

But the cscx has a -1 in front of it, and the 1/sinx has a +1

Answer 32

hmmm, a (+) on the right?

Answer 33

1/sinx +1

Answer 34

well it --> cot2^x/cscx+1 = 1-sinx/sinx <-- is what you had originally

Answer 35

so, can't say where the (+) came from

Answer 36

1-sinx/sinx
1/sinx - sinx/sinx

Answer 37

Ohhhhhh nevermind it's a -1

Answer 38

:D

Answer 39

cscx-1 = 1/sinx -1 Is correct so far, right?

Answer 40

yes

Answer 41

and of course from the "reciprocal identities", that'll turn into the same :)

Answer 42

and cscx = 1/sinx, so that's it?

Answer 43

yes

Answer 44

Thank you! :)

Answer 45

yw