## AmTran_Bus 2 years ago Prove this trig identity csc^4-csc^2=cot^4+cot^2

1. AmTran_Bus

|dw:1353025115000:dw|

2. lgbasallote

look at the LHS...try factoring out csc^2 what do you get?

3. AmTran_Bus

|dw:1353025247991:dw|

4. lgbasallote

not exactly....if you factor out csc^2 then csc^4 becomes csc^2 but csc^2 doesn't become csc^2

5. lgbasallote

|dw:1353025341525:dw|

6. AmTran_Bus

Would it even exist, that is, would it cancel? Or just be csc?

7. lgbasallote

what is csc^2/csc^2?

8. AmTran_Bus

ohhhh. I feel dumb. Got ya.

9. lgbasallote

so what should the factored form be?

10. AmTran_Bus

|dw:1353025535036:dw|

11. lgbasallote

right

12. lgbasallote

now what is csc^2 - 1?

13. AmTran_Bus

cot?

14. lgbasallote

cot^2

15. lgbasallote

so uou have $\huge \csc^2 (\cot^2)$ now here comes a tricky part...review your identities and give me the equation for csc^2

16. AmTran_Bus

csc2 = 1/sin...it also equals cot+1

17. lgbasallote

cot^2 + 1 not cot + 1

18. AmTran_Bus

19. lgbasallote

you should check your notes and make sure you wrote them right....you might get them wrong... $\huge \sin^2 + \cos^2 = 1$ $\huge \tan^2 + 1 = \sec^2$ $\huge \cot^2 + 1 = \csc^2$

20. lgbasallote

anyway...back to what i was saying... csc^2 is cot^2 + 1 so $\huge \csc^2 (\cot^2) \implies (\cot^2 + 1)(\cot^2)$ now expand that

21. AmTran_Bus

would the lhs say csc cot ^4? The RHS is cot^4 +cot^2

22. lgbasallote

i think you're confused...let me flip it... $\huge \cot^2 \theta (\cot^2\theta + 1)$ do you know how to distribute that now?

23. AmTran_Bus

Its not that I'm confused as much as tired my friend. |dw:1353026333309:dw|

24. lgbasallote

well anyway... cot^2(cot^2 + 1) becomes cot^4 + cot^2 so you have just proven the identity

25. AmTran_Bus

Thanks!!!!!!!

26. lgbasallote

welcome