115k views
1 vote
Use the double-angle formula for cosine to establish the identity cos (theta / 2) = ±cos(theta) + 1 / 2 .

User Patrickkx
by
7.0k points

1 Answer

2 votes

Answer:

See explanation and proof below.

Explanation:

For this case we want to proof this identity:


cos((\theta)/(2)) = \pm \sqrt{(1+ cos(x))/(2)}

And we need to us the double angle formula given by:


cos^2 (x) = (1+ cos (2x))/(2)

If we use a substitution for example
x = (\theta)/(2) we see that the double angle formila is given by:


cos^2((\theta)/(2)) = (1+ cos (2(\theta)/(2)))/(2)

And we got:


cos^2((\theta)/(2)) = (1+ cos (\theta))/(2)

And if we apply sqaure root on both sides we got:


cos((\theta)/(2)) = \pm \sqrt{(1+ cos (\theta))/(2)}

And that complete the proof

User Vetrivel Mp
by
7.5k points