# Thread: Proving a conditional indentity

1. ## Proving a conditional indentity

I need to prove a conditional identity cosA + cosB + cosC = 1 + 4sin(A/2) + sin(B/2) + sin(C/2)
given that A+B+C=pi. But the negative sign at the last is giving me trouble.

2. ## Re: Proving a conditional indentity

$\displaystyle \cos \left(\frac{A+B}{2}\right)$ is not equal to $\displaystyle -\cos\left(\frac{C}{2}\right).$

3. ## Re: Proving a conditional indentity

how?
cos[(A+B)/2]
=cos[(pi - c )/2]
= -cos (c/2)

4. ## Re: Proving a conditional indentity

$\displaystyle \frac{\pi - C}{2}=\frac{\pi}{2}-\frac{C}{2}.$

5. ## Re: Proving a conditional indentity

yes I know that but A+B+C=pi

so A+B=pi-c

6. ## Re: Proving a conditional indentity

!!!!!!!

Expand $\displaystyle \cos\left(\frac{\pi}{2}-\frac{C}{2}\right).$

7. ## Re: Proving a conditional indentity

i know that it is equal to sin(c/2) but i want to write it in the form of cos.

8. ## Re: Proving a conditional indentity

Well, sin(c/2) is certainly NOT equal to -cos(c/2)!

9. ## Re: Proving a conditional indentity

You can't without introducing a square root, instead replace the $\displaystyle \cos ^{2}\left(\frac{C}{2}\right)$ by $\displaystyle 1-\sin ^{2}\left(\frac{C}{2}\right).$

10. ## Re: Proving a conditional indentity

cos[(pi-c)/2]
=cos(-c/2)
=-cos(c/2)

at which step am I wrong?

11. ## Re: Proving a conditional indentity

The second line is wrong.
Read again posts 4 and 6.

12. ## Re: Proving a conditional indentity

Okay i got it.

thanks.