## Hannah_Ahn 3 years ago Prove the identity. $1 \over 1+\sin \theta$=$\sec ^{2} \theta$ -$\tan \theta \over \cos \theta$

1. nikvist

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

2. meverett04

Let us know if you need more help with this ...

3. myininaya

$\frac{1}{1+\sin(x)} \cdot \frac{1-\sin(x)}{1-\sin(x)}=\frac{1-\sin(x)}{1-\sin^2(x)}=\frac{1-\sin(x)}{\cos^2(x)}$ $=\frac{1}{\cos^2(x)}-\frac{\sin(x)}{\cos^2(x)}=\sec^2(x)-\frac{\sin(x)}{\cos(x)} \cdot \frac{1}{\cos(x)}$ $=\sec^2(x)-\tan(x) \cdot \frac{1}{\cos(x)}=\sec^2(x)-\frac{\tan(x)}{\cos(x)}$ but nikvist already did this i might in a little extra steps if you wanted to see those

4. Hannah_Ahn

why did you muliply 1 over cos theta?

5. Hannah_Ahn

it's so interesting how we can do that :p

6. myininaya

$\cos^2(x)=\cos(x) \cdot \cos(x)$

7. Hannah_Ahn

oh i missed that. thanks guys!!!! :)

8. meverett04

I don't think they multiplied 1 over cos theta .... I think they used the Pythagorean Identity sin^2 x + cos^2 x = 1 or 1 - sin^2 x = cos^2 x You will use this formula many many many times in trig

9. Hannah_Ahn

do you guys memorize these formulas?

10. myininaya

Yes!

11. Hannah_Ahn

I guess I have to do lots of these questions to master this chapter..

12. Hannah_Ahn

tehehehehe aywy thanks so much lifesaverss :)

13. meverett04

I only memorize the ones I see over and over again. For trig, there are just a few that I memorize sin^2 x + cos^2 x = 1 is one of them, but I have used it so many times, it is no longer memorized, I know it like my name meverett lol

14. Hannah_Ahn

I am having troubles with every single questions.. wowo.... how do i prove this one.? cot theta over csc theta - 1 equals csc theta + 1 over cot theta ???

15. Hannah_Ahn

haha I don't even see the first step...

16. meverett04

For cot x and csc x I would change them into sines and cosines, can you do this?

17. Hannah_Ahn

......?

18. Hannah_Ahn

for csc x 1 over sin x

19. meverett04

No .... for cot theta over csc theta

20. TuringTest

is it${\cot\theta\over\csc\theta-1}={\csc\theta+1\over\cot\theta}$??

21. Hannah_Ahn

yes

22. meverett04

I would change cot (theta) = cos(theta) / sin (theta) csc(theta) = 1/sin(theta) for all cot and csc

23. Hannah_Ahn

yes I did that just now :)

24. TuringTest

Yes meverett is right, though this one is a little tricky. I had to change both sides.

25. Hannah_Ahn

and multiply each side by 1 over sin theta - 1?

26. TuringTest

sorry no... let me fix that.

27. TuringTest

${{\cos x\over\sin x}\over{1\over\sin x}-1}={{{1\over\sin x}+1}\over{\cos x\over\sin x}}$right?

28. Hannah_Ahn

uh... i don't get the 2nd part. you flip them?

29. TuringTest

no, that's just the right side of the formula with csc written as 1/sin, and cot written as cos/sin.

30. Hannah_Ahn

oh hahahaha gotcha

31. TuringTest

I changed both sides to their elementary functions is all. I think there are too many sines in this equation; makes my head hurt. But it looks like if we multiply each side by$\frac{\sin x}{\sin x}$everything will be a bit simpler, so try that. See if you can write what you get.

32. Hannah_Ahn

uhh I am sorry I don't want your head to hurt.. but I don't know how to do it.. :S

33. TuringTest

no prob, here it is...

34. TuringTest

$\frac{\frac{\cos x}{\sin x}}{\frac{1}{\sin x}-1}\frac{\sin x}{\sin x}=\frac{\frac{1}{\sin x}+1}{\frac{\cos x}{\sin x}}\frac{\sin x}{\sin x}$$\frac{\cos x}{1-\sin x}=\frac{1+\sin x}{\cos x}$reasonable?

35. TuringTest

we just multiplied the top and bottom of each fraction by sinx

36. Hannah_Ahn

yes!

37. TuringTest

Ok, so now I'm going to say that my mission is to get the left to look like the right. I'm not going to change the right anymore, just try to get the left to look like it. So see if you can multiply the left by (1+sinx)/(1+sinx). That should make you see something.

38. TuringTest

it should change the denominator in a familiar way...

39. Hannah_Ahn

i got it 'till $cosx + sinxcosx \over 1-\sin ^{2} x$

40. TuringTest

That's good. You didn't need to multiply out the top, you could have left it as cosx(1+sinx). You'll see why in this step. Look at the denominator. What is that identity?

41. Hannah_Ahn

$\cos ^{2}x$

42. TuringTest

so we can divide out our fraction by a cosx to make it...?

43. Hannah_Ahn

yes!!!!!!!!!!!!!!!!!!!!!!!!!!

44. TuringTest

there ya go :)

45. Hannah_Ahn

THANKS! yayayayayay!

46. Hannah_Ahn

you are really qualified as an awesome tutor! so patient.. :') i wish you the bestest luck !

47. myininaya

Turing is awesome!

48. TuringTest

so what was the strategy? 1)convert all tan, cot, sec, and csc into terms of sin and cos. 2)Simplify if possible. Remember that mertsj was right, you cannot "divide both sides" or "multiply both sides" in identities. You need to work each side separately. 3)Look for ways make things like (1-sin^2x) so we can change it to cos^2x. That was all my strategy was, plus practice. Happy New Year!

49. Hannah_Ahn

so are you though, your every each step helped me alot :)

50. Hannah_Ahn

myinininaya :)

51. TuringTest

Yay myin!

52. Hannah_Ahn

LOVELY! happy NEW year :) bye ttyl

53. AravindG

nikvist can u hlp me??