## anonymous one year ago VERIFY THE IDENTITY: 1/csc(x-1) + 1/csc(x+1)=2/csc(x)-sin(x)

1. freckles

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

2. anonymous

close, only it says 2/csc(x)-sin(x) not sure if that is the same thing or not?

3. freckles

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

4. freckles

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

5. anonymous

yes

6. freckles

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

7. anonymous

yes that is correct

8. freckles

that doesn't seem like an identity

9. freckles

try inputting pi/6

10. freckles

both sides aren't the same

11. anonymous

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

12. freckles

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

13. anonymous

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

14. freckles

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

15. freckles

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

16. anonymous

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

17. freckles

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

18. freckles

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

19. anonymous

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

20. UnkleRhaukus

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

21. anonymous

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

22. UnkleRhaukus

$\frac{ 2 }{ \csc x-\tfrac1{\csc(x)}}$

23. anonymous

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

24. UnkleRhaukus

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

25. anonymous

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