Question

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

Answer 1

Answer:

See explanation and proof below.

Step-by-step explanation:

For this case we want to proof this identity:

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

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

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

If we use a substitution for example $x = \frac{\theta}{2}$ we see that the double angle formila is given by:

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

And we got:

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

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

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

And that complete the proof