## Babynini one year ago Prove the identity (tanx)/(1-cosx) = cscx(1+secx)

1. Babynini

I've gotten up to (sinx/cosx)(1/(1-cosx)

2. Babynini

is that correct so far?

3. freckles

ok and you can write that as:$\frac{\frac{\sin(x)}{\cos(x)}}{1-\cos(x)} \\ \frac{\sin(x)}{\cos(x)} \cdot \frac{1}{1-\cos(x)}$

4. freckles

oh that is what you wrote

5. freckles

lol

6. freckles

$\frac{\sin(x)}{\cos(x)} \cdot \frac{1}{1-\cos(x)} \cdot \frac{1+\cos(x)}{1+\cos(x)}$ see what this does maybe

7. freckles

multiply the second and last fraction

8. Babynini

so the last fraction is just like multiplying by 1 right?

9. freckles

yep

10. Babynini

It'd be (sinx/cosx)((1+cosx)/(1-cos^2x))

11. freckles

right and 1-cos^2(x) is sin^2(x)

12. freckles

$\frac{\sin(x)}{\cos(x)} \cdot \frac{1+\cos(x)}{\sin^2(x)} \\ \frac{\sin(x)}{\sin^2(x)} \frac{1+\cos(x)}{\cos(x)}$

13. freckles

can you see where to go from here?

14. Babynini

Emm no haha sorry.

15. freckles

$\frac{\sin(x)}{\sin^2(x)}(\frac{1}{\cos(x)}+\frac{\cos(x)}{\cos(x)})$ how about here?

16. freckles

cos(x)/cos(x)=? 1/cos(x)=? sin(x)/sin^2(x)=?

17. Babynini

1) = 1 2) = secx 3) I'm not sure about that one :P

18. Babynini

where did you get the cosx/cosx?

19. freckles

1) good 2) good 3) do you know how to simplify say something like u/u^2 ? separated the fraction to "where did you get the cosx/cosx? " $\frac{a+b}{c}=\frac{a}{c}+\frac{b}{c} \\ \text{ we had } \frac{1+\cos(x)}{\cos(x)}=\frac{1}{\cos(x)}+\frac{\cos(x)}{\cos(x)}$

20. Babynini

aah I see, ok :)

21. Babynini

3) um, i'm not sure.

22. freckles

$\frac{u}{u^2}=\frac{u}{u \cdot u}=\frac{\cancel{u}}{\cancel{u} u}=\frac{1}{u} \\ \text{ or just use law of exponents } \\ \frac{u}{u^2}=u^{1-2}=u^{-1}=\frac{1}{u} \\ \frac{\sin(x)}{\sin^2(x)}=\frac{1}{\sin(x)}$

23. freckles

and 1/sin(x) is....

24. Babynini

cscx!

25. freckles

yep

26. freckles

What we did: $\frac{\tan(x)}{1-\cos(x)} \\ \tan(x) \cdot \frac{1}{1-\cos(x)} \\ \frac{\sin(x)}{\cos(x)} \frac{1}{1-\cos(x)} \\ \frac{\sin(x)}{\cos(x)} (\frac{1}{1-\cos(x) } \cdot 1) \\ \frac{\sin(x)}{\cos(x)} (\frac{1}{1-\cos(x) } \cdot \frac{1+\cos(x)}{1+\cos(x)} ) \\ \frac{\sin(x)}{\cos(x)}(\frac{1+\cos(x)}{1-\cos^2(x)}) \\ \frac{\sin(x)}{\cos(x)} \frac{1+\cos(x)}{\sin^2(x)} \\ \frac{\sin(x)}{\sin^2(x)} \frac{1+\cos(x)}{\cos(x)} \\ \frac{1}{\sin(x)} \frac{1+\cos(x)}{\cos(x)} \\ \frac{1}{\sin(x)} (\frac{1}{\cos(x)}+\frac{\cos(x)}{\cos(x)}) \\ \csc(x)(\sec(x)+1) \\ \csc(x)(1+\sec(x))$

27. Babynini

aw thanks for taking the time to type that all up!

28. Babynini

haha I guess I should do a few more practice ones. Sigh. Thanks so much for the help :)

29. freckles

np trig identities can be tricky (Takes practice practice) but fun just think you are doing a puzzle or something puzzles should be fun

30. Babynini

Tricky may be an understatement xP haha sure sure.