## anonymous one year ago how do i verify (cos x/ 1 + sin x) + (1 + sin x/ cos x) = 2 sec x? i don't know how to verify anything at all can someone explain how it works?

1. Nnesha

$\huge\rm \frac{ \cos(x) }{ 1+\sin(x) } +\frac{ 1+\sin(x) }{ \cos(x) }$ 1)what is the common denominator ?

2. anonymous

i dont know

3. anonymous

4. Nnesha

alright let's change that to simple algebra

5. Nnesha

$\huge\rm \frac{ x }{ y }+\frac{ y }{ x }$ what's the common denominator ?

6. anonymous

xy

7. Nnesha

okay let's say y represent 1+sinx and x = cos(x) $y=1+\sin(x)$ $x=\cos(x)$ now can you tell me what would be the common denominator$\huge\rm \frac{ \cos(x) }{ 1+\sin(x) } +\frac{ 1+\sin(x) }{ \cos(x) }$ for this question :=)

8. anonymous

1+sinx*cosx ?

9. Nnesha

PERFECT!

10. anonymous

hi i speak Arabic

11. anonymous

salam alakom

12. anonymous

Nnesha

13. Nnesha

$\huge\rm \frac{ \cos(x) }{ \color{red}{1+\sin(x)} } +\frac{ 1+\sin(x) }{\color{blue}{ \cos(x)} }$ $\huge\rm \frac{ \cos(\color{blue}{cos}) + (\color{reD}{1+\sin})(1+\sin)}{ \cos(x) (1+\sin(x))}$ multiply the numerator of 1st fraction by the denominator of 2nd fraction multiply the numerator of 2nd fraction with the denominator of 1st fraction

14. anonymous

so would it become cos^2x and (1+sin)^2 on the numerator??

15. Nnesha

yes right!!!

16. Nnesha

$(1+\sin(x))^2 = (1+\sin)(1+\sin)$ (1+sin)^2 is same as (1+sin)(1+sin) foil it

17. anonymous

and what happens when i foil it?

18. Nnesha

you will get 3 terms :3

19. Nnesha

then you will be able to cancel out some stuff at the numerator :=)

20. anonymous

so 1+2sinx+sin^2x ???

21. Nnesha

yes right $\huge\rm \frac{ \color{ReD}{cos^2x}+1+2sinx+\sin^2x }{ cosx(1+sinx) }$ cos^2x= what ? familiar with the identity ?

22. Nnesha

$\huge\rm sin^2 \theta+\cos^2\theta=1$ this one solve for cos^2

23. anonymous

cos^2=1-sin^2 ?

24. Nnesha

replace cos^2 with 1-sin^2 and then simplify !

25. Nnesha

$\huge\rm \frac{ \color{ReD}{1-sin^2x}+1+2sinx+\sin^2x }{ cosx(1+sinx) }$

26. anonymous

2+2sinx ?

27. Nnesha

yes! what is common factor ?

28. Nnesha

$\huge\rm \frac{ 2+2sinx }{ \cos(x)(1+sinx) }$ 2+sinx what is common in both terms?

29. anonymous

sinx?

30. Nnesha

no first term is 2 and 2nd one is 2sinx so sinx isn't common

31. anonymous

so 2

32. Nnesha

yes right take out the 2 what will you get ?

33. anonymous

sinx

34. Nnesha

nah

35. Nnesha

when you take out 2 from 2 what will you have left wth?

36. anonymous

0

37. Nnesha

haha well no in other words you have to divide both terms with the common factor so $\huge\rm \frac{ 2}{ 2} +\frac{ 2sinx }{ 2 } =2(??+???)$

38. anonymous

1+sinx ?

39. Nnesha

yes! and we should keep the common factor outside the parentheses $\huge\rm \frac{ 2(1+sinx) }{ cosx(1+sinx) }$

40. Nnesha

simplify! that

41. anonymous

2/cosx

42. Nnesha

yes right and last step ?

43. anonymous

i dont know

44. Nnesha

thats fine! 2/cosx is same as $\huge\rm 2 \times \frac{ 1 }{ cosx }$

45. Nnesha

and 1/cosx= ??

46. anonymous

secx

47. Nnesha

so 2/cosx= ? :=)

48. anonymous

EXCITING

49. anonymous

i honestly have no idea how to do anything with identities so thank you

50. Nnesha

ikr! trig is FUN(well i hate graph)

51. Nnesha

my pleasure! :=) welcome to openstudy!

52. Nnesha

$$\color{blue}{\text{Originally Posted by}}$$ @rami2021 salam alakom $$\color{blue}{\text{End of Quote}}$$ Walikum salam and yeah arabic is my fvt language but all i know is *kafya halik ?? ;~;

53. Nnesha

54. anonymous

no problem!

55. anonymous

i'm about to post another question, i have one left