## anonymous one year ago prove the following trigonometric identities. 2/(rt(3)cos(x)+sin(x))=sec(pi/6-x) tan(x/2)=sin(x)/(1+cos(x))

1. anonymous

@dinamix

2. anonymous

$\frac{ 2 }{\sqrt{3} \cos(x)+\sin(x)}=\sec(\pi/6-x) \tan(x/2)=\frac{ \sin (x) }{1+\cos (x) }$ is that your question?

3. anonymous

no they are two separate equations. 2/(rt(3)cos(x)+sin(x))=sec(pi/6-x) tan(x/2)=sin(x)/(1+cos(x))

4. anonymous

oK

5. anonymous

I will do only one, which one do you like?

6. anonymous

do the top one

7. anonymous

you can do either one if you want

8. IrishBoy123

for the second $tan\frac{x}{2}=\frac{sinx}{1+cosx}$ try this $tan\frac{x}{2}=\frac{2sin\frac{x}{2}cos \frac{x}{2}}{1+(1-2sin^2 \frac{x}{2})}$

9. welshfella

sec (pi/6 - x) = 1 / (sqrt3/2 cos x + sin pi/6 sin x) = 2 / (sqrt 3 cos + sin x)

10. welshfella

* I missed out the second step which is = 1 / (cos pi/6 cos x + sin pi/6 sin x)

11. anonymous

$\frac{ 2 }{\sqrt{3} \cos(x)+\sin(x)}=\sec(\pi/6-x)$ $\frac{ 2 }{\sqrt{3} \cos(x)+\sin(x)}=\frac{ 1 }{ \cos{(\pi/6-x)} }$ using:cos(u-v)=cos u cos v+sin u sin v $\frac{ 2 }{\sqrt{3} \cos(x)+\sin(x)}=\frac{ 1 }{ \cos{(\pi/6) \times \cos(x)}+\sin{(\pi/6) \times \sin(x)} }$

12. anonymous

i got that far ASAAD123 put i didn't know what to do after that

13. welshfella

the denominator = cos (pi/6 - x)

14. welshfella

and 1 / cos = sec

15. anonymous

wouldn't you also change sin(pi/6) to 1/2

16. welshfella

no its the sin of a compound angle (pi/6 - x) not sin pi/6

17. anonymous

ok

18. anonymous

19. anonymous

so how do you get the sides to equal

20. welshfella

i had to use sin pi/6 = 1/2 because i started with RHS and converted to LHS whereas ASAAD did the reverse

21. welshfella

with these identities you choose one side and try to convert it to the other. This proves the identity.

22. anonymous

ok

23. anonymous

i think i understand it now

24. welshfella

so you can use asaad's or mine. Either would do.

25. welshfella

There is only more more step to prove IrishBoy's solution

26. anonymous

i already got irish boy's solution