## anonymous one year ago Verify the identity. ((cos x)/1+sin x) + ((1+sinx)/cos x) =2secx

1. anonymous

$\frac{ cosx }{ 1+\sin x }+\frac{ 1+\sin x }{ \cos x }=2secx$

2. anonymous

I don't know if I'm doing this right. I used LCD.

3. anonymous

$LEFT SIDE: \frac{ \cos^2x +(1+\sin x)(1+\sin x)}{ \cos(1+sinx) }$

4. anonymous

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

5. Mertsj

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

6. Nnesha

change cos^2 by 1-sin^2x identity

7. anonymous

Wait, am I doing it right?

8. Mertsj

$\frac{2\cos x}{\cos ^2x}=\frac{2}{\cos x}=2\sec x$

9. anonymous

@Mertsj What did you do first?

10. Mertsj

Multiply first fraction by (1-sinx)/(1-sinx)

11. Mertsj

Multiply second fraction by cosx/cosx

12. anonymous

Ohhh is it like multiplying the conjugate?

13. Mertsj

it's really setting up cos^2x in both denominators

14. Mertsj

Because 1-sin^2x=cos^2x

15. Nnesha

$\huge\rm \frac{ \color{Red}{\cos^2x} +1 +2sinx + \sin^2 }{ cosx(1+sinx) }$ $\large\rm \frac{ \color{Red}{1-sin^2x} +1 +2sinx + \sin^2 }{ cosx(1+sinx) }$ sin^2 cancel each other out you will get $\large\rm \frac{ \color{Red}{1\cancel{-sin^2x}} +1 +2sinx +\cancel{ \sin^2} }{ cosx(1+sinx) }$

16. anonymous

Ohhh so what I was doing is wrong. Haha! Thanks!

17. Nnesha

i don't think so..

18. Mertsj

Not wrong, such not very effective.l

19. anonymous

Like it the solution would be longer? :D

20. Nnesha

$\huge\rm \frac{ 2+2sinx }{ cosx(1+sinx) }$ take out 2 $\rm \frac{ 2(1+sinx) }{ cosx(1+sinx)}$

21. anonymous

There's no "it" in my last sentence. Haha.

22. Mertsj

Typically there are a variety of ways to solve these identities.

23. anonymous

Ohhhh @Nnesha thank you so much! :) You did the whole solution for me.

24. Nnesha

true but i think the way you were doing is easy :D

25. anonymous

That's why this is harder because there are so many ways to do it and it's so frustrating haha!

26. Nnesha

and like mertsj said there are more than 2 ways to verify the identities