## Ammarah one year ago Identities HELP FINAL TOMORROW

1. Ammarah

|dw:1433373713959:dw|

2. Ammarah

@perl

3. perl

4. Ammarah

yess

5. perl

$$\Large { \tan x ( 1 + \cos(2x) )= \\ = \tan x ( 1 + 2\cos^2(x)-1 )= }$$

6. perl

i used the substitution cos(2x) = 2cos^2 x -1

7. Ammarah

how did u get rid of the cosx>

8. perl

$$\Large { \tan x ( 1 + \cos(2x) ) \\ = \tan x ( 1 + \color{red}{ 2\cos^2(x)-1} ) \\= \tan x \cdot 2\cos^2(x) \\= \frac{\sin x}{\cos x } \cdot 2\cos^2(x) \\~\\= 2 \sin x \cos x \\~\\= \color{blue}{\sin(2x)} }$$

9. perl

the red and blue are substitutions

10. perl

Trig identities: sin(2x) = 2sin x cos x cos(2x) = 2cos^2 x - 1

11. Ammarah

i know but u know u cosx was left

12. Ammarah

how did u get rid of that..

13. perl

in the last line, 2sin x cos x = sin(2x) , thats a trig identity

14. perl

You can prove that by use of the addition formula: sin(2x) = sin(x+x) = sin x cos x + cos x sin x = 2 sin x cos x

15. perl

this is a good resource for identities http://www.purplemath.com/modules/idents.htm

16. Ammarah

ohkk

17. Ammarah

i have another queestion

18. Ammarah

|dw:1433375868645:dw|

19. perl

Is this the question? $$\Large \csc x = \frac{\cot ( x/2)}{1 + \cos x }$$

20. Ammarah

yes

21. perl

Let's use the half angle identity.

22. perl

tan (x/2) = sin x / ( 1 + cos x )

23. perl

cot(x/2) = 1 / tan(x/2) = ( 1 + cos x ) / sin x

24. perl

thats an equivalent identity

25. perl

cot(x/2) / ( 1 + cos x ) =cot(x/2) * 1 / ( 1 + cos x ) = (1 + cos x ) / sin x * 1 / ( 1 + cos x ) = 1 / sin x = csc x

26. perl

I used this identity http://prntscr.com/7cuud8

27. Ammarah

ok thanks