## anonymous one year ago HELP NEEDED!!! WILL FAN AND MEDAL!!!! I need help understanding how to prove identities.

1. anonymous

2. anonymous

what is it?

3. anonymous

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

4. anonymous

It was actually supposed to be tangent theta times cosine theta over one plus sine theta

5. anonymous

I have all of the identities written out so I can easily refer to them, but the problem I am having is learning how to combine the like terms.

6. anonymous

$\frac{ \sin \theta }{ \cos \theta }+\frac{ \cos \theta }{ 1+ \sin \theta }$ this is what I need help combining because I have already converted Tan

7. anonymous

what the heck is the tan doinog there ?

8. anonymous

The tan was there because our teacher was trying to get us to recreate all of the identities to where they equal one another

9. anonymous

no don't convert to tangent in fact, if it was tangent, you would convert to $$\frac{\sin(x)}{\cos(x)}$$

10. anonymous

lets do a tiny bit of algebra first

11. anonymous

That's what I did and that was my farthest point in this.

12. anonymous

ok i got that wrong, hold the phone

13. anonymous

$\frac{b}{a}+\frac{a}{1+b}$ that's better can you add these?

14. anonymous

I have no idea how to... or at least I can't remember. I know with multiplying you foil it, if you divide you use the kcf format :/ but adding and subtracting makes no sense to me because all I know is a common denominator is required

15. anonymous

lol that is always the problem, the algebra

16. anonymous

here is the one true way to add fractions $\huge \frac{A}{B}+\frac{C}{D}=\frac{AD+BC}{BD}$

17. anonymous

o.o"

18. anonymous

in your case you will have $\frac{b(1+b)+a^2}{a(1+b)}$

19. anonymous

I am getting more lost by the minute...

20. anonymous

that is the way to add fractions if they are numbers$\frac{2}{7}+\frac{3}{5}=\frac{2\times 5+7\times 3}{7\times 5}$

21. anonymous

and it is also the way to add fractions if they have variables it is the only way to do this, by using algebra to add

22. anonymous

ohhhh ok, I get it, sort of.

23. anonymous

forget that "least common denominator" nonsense you were taught that is the way to add, you cannot avoid it

24. anonymous

25. anonymous

Yeah I suppose

26. anonymous

using the one true way to add $\frac{b}{a}+\frac{a}{1+b}$ you get $\frac{b(1+b)+a^2}{a(1+b)}$

27. anonymous

now some more algebra, but just a little $\frac{b(1+b)+a^2}{a(1+b)}=\frac{b+b^2+a^2}{a(1+b)}$

28. anonymous

that is just the distributive law in the numerator now that we are done with algebra, we can go back to sines and cosines

29. anonymous

sooooooo then $\frac{ \cos \theta }{ \sin \theta } + \frac{ \cos \theta }{ 1+\sin \theta }$ should become....

30. anonymous

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

31. anonymous

Ok. But now comes an even trickier part.... I have to find a way to reduce that

32. anonymous

hold a sec the initial question was $\frac{ \sin \theta }{ \cos \theta }+\frac{ \cos \theta }{ 1+ \sin \theta }$ right?

33. anonymous

Yes

34. anonymous

so after the algebra we get to $\frac{\sin(x)+\sin^2(x)+\cos^2(x)}{\cos(x)(1+\sin(x))}$ don't look to reduce yet

35. anonymous

the usual miracle occurs, $\sin^2(x)+\cos^2(x)=1$ the mother of all trig identities

36. anonymous

correct

37. anonymous

The reasoning behind adding and subtracting the fractions is exactly what was said above, but the step being left out from above that might make it clear is that you just multiply both fractions by a number that will make both denominators the same or "common". In the above example....$\frac{ b }{ a }+\frac{ a }{ 1+b } = (\frac{ 1+b }{ 1+b })(\frac{ b }{ a })+(\frac{ a }{ a })(\frac{ a }{ 1+b })$ and that is where the other equation @satellite73 came up with came from. Not to inturrupt

38. anonymous

$\frac{\sin(x)+\overbrace{\sin^2(x)+\cos^2(x)}^{\text{ this is 1}}}{\cos(x)(1+\sin(x))}$

39. anonymous

$\frac{\sin(x)+1}{\cos(x)(1+\sin(x))}$ NOW you can cancel the common factor of $1+\sin(x)$ top and bottom

40. anonymous

Ok, ok, I see it now. so the Sins actually match and leave you with $\frac{ 1 }{ \cos \theta }$

41. anonymous

yes if you want to impress your teacher, write it as $\sec(\theta)$

42. anonymous

i hope you also see that 98% of this is algebra

43. anonymous

which, if your algebra is not what it needs to be, is going to be a problem, so bone up on it also use the gimmick of replacing sine and cosine by letters to make the algebra easier if we had to do this writing sine and cosine each time it would have been a pain