Question

VERIFY THE IDENTITY:
1/csc(x-1) + 1/csc(x+1)=2/csc(x)-sin(x)

Answer 1

question is it really:
\[\frac{1}{\csc(x-1)}+\frac{1}{\csc(x+1)}=\frac{2}{\csc(x)}-\sin(x)\]

Answer 2

close, only it says 2/csc(x)-sin(x)

not sure if that is the same thing or not?

Answer 3

isn't that what I typed for the right hand side?

Answer 4

did you mean to write 2/(csc(x)-sin(x)) instead?

Answer 5

yes

Answer 6

\[\frac{1}{\csc(x-1)}+\frac{1}{\csc(x+1)}=\frac{2}{\csc(x)-\sin(x)}\]

Answer 7

yes that is correct

Answer 8

that doesn't seem like an identity

Answer 9

try inputting pi/6

Answer 10

both sides aren't the same

Answer 11

Ok that is what I kept coming up with, I just wanted to verify that it was not an identity, thank you!

Answer 12

did you mean to write
\[\frac{1}{\csc(x)-1}+\frac{1}{\csc(x)+1}=\frac{2}{\csc(x)-\sin(x)}\]
this is an identity

Answer 13

I cannot figure out how it is an identity, however....

Answer 14

so is that not what you are trying to prove is an identity ? it really is the first equation you wrote?

Answer 15

If you really meant to write that last thing I wrote, try combining the fractions on the left hand side first

Answer 16

I think I am typing it wrong, this is exactly how it is written on the worksheet: \[\frac{ 1 }{ \csc x-1 }+\frac{ 1 }{ \csc x+1 }=\frac{ 2 }{ \csc x-\sin x }\]

Answer 17

then all you have to do is divide both top and bottom by csc(x)

Answer 18

2 steps really:
combine fractions
divide top and bot by csc(x)

Answer 19

How can you combine the fractions when the denominator are not the same? I am not sure how to make them the same

Answer 20

\[LHS=\frac{1}{\csc(x)-1}+\frac{1}{\csc(x)+1}\\
=\frac{1}{\csc(x)-1}\times\frac{\csc(x)+1}{\csc(x)+1}+\frac{1}{\csc(x)+1}\times\frac{\csc(x)-1}{\csc(x)-1}\\
=\frac{2\csc(x)}{\csc^2(x)-1}\\
=\]
now divide both numerator and denominator by csc

Answer 21

Like this? \[\frac{ 2 }{ \csc x-1}\]

Answer 22

\[\frac{ 2 }{ \csc x-\tfrac1{\csc(x)}}\]

Answer 23

ohh of course. I understand now. Thank you so much!

Answer 24

all clear?

do you understand how we combined those fractions at the start?

Answer 25

yes, I kept trying to do it that way but for some reason I was doing the algebra wrong....I need to practice more