How do you simplify ${cos}^{2} (\frac{θ}{2}) - {sin}^{2} (\frac{θ}{2})$ $cos^2 (theta/2) − sin^2 (theta/2)$ ?

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How do you simplify ${cos}^{2} (\frac{θ}{2}) - {sin}^{2} (\frac{θ}{2})$ $cos^2 (theta/2) − sin^2 (theta/2)$ ?

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Answer 1 · 2016-05-17T20:28:31

$cos θ$ $costheta$

Explanation:

We will use the following identities:

$cos (\frac{θ}{2}) = \sqrt{\frac{1 + cos θ}{2}}$ $cos(theta/2)=sqrt((1+costheta)/2)$
$sin (\frac{θ}{2}) = \sqrt{\frac{1 - cos θ}{2}}$ $sin(theta/2)=sqrt((1-costheta)/2)$

Thus, substituting these into the expression, we get:

${cos}^{2} (\frac{θ}{2}) - {sin}^{2} (\frac{θ}{2}) = {(\sqrt{\frac{1 + cos θ}{2}})}^{2} - {(\sqrt{\frac{1 - cos θ}{2}})}^{2}$ $cos^2(theta/2)-sin^2(theta/2)=(sqrt((1+costheta)/2))^2-(sqrt((1-costheta)/2))^2$

$= \frac{1 + cos θ}{2} - \frac{1 - cos θ}{2}$ $=(1+costheta)/2-(1-costheta)/2$

$= \frac{1 + cos θ - (1 - cos θ)}{2}$ $=(1+costheta-(1-costheta))/2$

$= \frac{1 - 1 + cos θ + cos θ}{2}$ $=(1-1+costheta+costheta)/2$

$= \frac{2 cos θ}{2}$ $=(2costheta)/2$

$= cos θ$ $=costheta$

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How do you simplify ${cos}^{2} (\frac{θ}{2}) - {sin}^{2} (\frac{θ}{2})$ $cos^2 (theta/2) − sin^2 (theta/2)$ ?

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