Hannah_Ahn 3 years ago prove the identity csc theta over tan theta + cot theta = cos theta

1. Zed

$\frac{ \frac{1}{sin {\theta}}}{tan{\theta}}+\frac{1}{tan{\theta}}=cos{\theta}$

2. Hannah_Ahn

I am sorry, how did that happen?

3. Zed

Sorry I will write it up first and post it with steps. Just a moment

4. Hannah_Ahn

Sweet, thanks!

5. Yifan12879

does it mean that (1/sin x)^2+(1/sin x)=1?

6. Yifan12879

if you divide the cos theta from both sides

7. Zed

Yes correct

8. Hannah_Ahn

I still would appreciate to see the steps plz :s

9. Yifan12879

but why? let x=pi/6 then 4+2=1??

10. Zed

yes I'm having that issue too. Are you sure it's written down right?

11. Hannah_Ahn

$\csc \theta \over \tan \theta +\cot \theta$ = [\cos \theta\]

12. Zed

I see now, next time put the tan theta + cot theta in brackets to make it clearer.

13. Yifan12879

multiply the LHS by sinx cosx

14. Hannah_Ahn

15. Yifan12879

i mean csc x /(tan x + cot x)= cscx sinx cosx / (tanx sinx cosx+cotx sinx cosx)=cosx/(sin^2 x+cos^2 x)=cosx

16. Yifan12879

im sorry it should be theta...

17. Hannah_Ahn

why did you multiply each sides by sinx cosx? how could u tell?

18. Zed

$\frac{1/sin{\theta}}{tan{\theta}+1/tan{\theta}}=cos{\theta}$$1/sin{\theta}=tan{\theta}cos{\theta}+cos{\theta}/tan{\theta}$$1/sin{\theta}=sin{\theta}+cos^2{\theta}/sin{\theta}$$1/sin{\theta}=sin^2{\theta}/sin{\theta}+cos^2{\theta}/sin{\theta}$$1/sin{\theta}=1/sin{\theta}$$1=1$

19. Zed

Here's a different way to go about it :)

20. Yifan12879

because there are sinx cosx in the denominator

21. Hannah_Ahn

my teacher doesn't let me combite the left side and the right side.. in this case i would need to make the left side cos theta. :S

22. Hannah_Ahn

zed; yur way is something to think about, and it's cool. yet i won't b able to use your method

23. Yifan12879

@Zed you shouldnt give out a prove like this, you should write LHS=...=....=...=RHS

24. Hannah_Ahn

yifian; i don't really get how you got there.. :(

25. Hannah_Ahn

i can't figure it out

26. Yifan12879

which step?

27. Hannah_Ahn

is there other way to do it?

28. Zed

Thanks Yifan, I haven't done proof in a long time. $LHS = \frac{1/sinx}{sinx/cosx+cosx/sinx}$ $=\frac{1/sinx}{(sin^2x+cos^2x)/sinxcosx}$$=\frac{1/sinx}{1/sinxcosx}$$=\frac{1}{cosx}$$=RHS$

29. Zed

sorry that second last line was supposed to be cosx

30. Yifan12879

yeah that is a good prove except that you substitute theta for x..

31. Zed

yeah i got tired of writing theta out

32. Zed

thanks Yifan

33. Hannah_Ahn

1 over cos x isn't same as cos x

34. Zed

It was a typo

35. Hannah_Ahn

i got that too.

36. Hannah_Ahn

well thanks for your time :)

37. Hannah_Ahn

i still don't get it . for some reason, im very confused. i will ask my teacher and I will let you know if there is a simpler way to do it if you are interested :P g'nite.