## anonymous 5 years ago I'm working on trig identities and in need of some help : (1-sec x)/(1+sec x)= (cos x-1)/(cos x+1) AND (cos x)/(1+sin x) + (1+sin x)/ (cos x)= 2 sec x

1. anonymous

Your first one can be solved by the definition of secant:$\frac{1-\sec x}{1+ \sec x}=\frac{1-\frac{1}{\cos x}}{1+\frac{1}{\cos x}}$

2. anonymous

$=\frac{\frac{\cos x-1}{\cos x}}{\frac{\cos x +1}{\cos x}}=\frac{\cos x -1}{\cos x +1}$

3. anonymous

(1-1/cosx)/(1+1/cosx) =(cosx-1)/cosx/(cosx+1)/cosx =(cosx-1)(cosx+1)

4. anonymous

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

5. anonymous

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

6. anonymous

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

7. anonymous

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

8. anonymous

What are the identities used to get from this one fraction to the next? cos x +1+2sin x+ sin ^{2} x div cos x times 1 + sin x = 2+ 2 sin x div cos x times 1 + sin x

9. anonymous

I thought that's what you were writing about...$\frac{\cos^2x+1+2\sin x+\sin^2x}{\cos x(1+\sin x)}$$=\frac{(\cos^2 x+ \sin^2 x)+1+2\sin x}{\cos x (1+\sin x)}$$=\frac{1+1+2\sin x}{\cos x(1+\sin x)}=\frac{2+2\sin x}{\cos x(1 + \sin x)}=\frac{2(1+\sin x)}{\cos x(1+\sin x)}$

10. anonymous

It's just the $\cos^2x + \sin^2 x =1$ identity.

11. anonymous

Both of you, thank you so very much for your expertise with this problems, it helps me greatly!!!! I really appreciate it! :)

12. anonymous

You're welcome :) Thanks for the appreciation - not everyone bothers to say thank you!

13. anonymous

I mean it, I am very greatful for the help, thanks for taking the time to help me, you deserve a thank you, you both do :)

14. anonymous

my pleasure;-) @lokisan i bother if my network doesnot fail;-)

15. anonymous

my pleasure;-) @lokisan i bother if my network doesnot fail;-)