Question #16d46

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Question #16d46

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Answer 1 · 2017-04-17T04:25:39

$tan (\frac{x}{2}) = \sqrt{\frac{1 - cos x}{1 + cos x}}$ $tan (x/2) = sqrt((1 - cos x)/(1 + cos x)$

Explanation:

Use trig identities:
$2 {sin}^{2} x = 1 - cos 2 x$ $2sin^2x = 1 - cos 2x$
$2 {cos}^{2} x = 1 + cos 2 x$ $2cos^2 x = 1 + cos 2x$
In this case:
$tan (\frac{x}{2}) = \frac{sin (\frac{x}{2})}{cos (\frac{x}{2})}$ $tan (x/2) = sin (x/2)/(cos (x/2)$
Find $sin (\frac{x}{2})$ $sin (x/2)$ and $cos (\frac{x}{2})$ $cos (x/2)$ in terms of cos x.
${sin}^{2} (\frac{x}{2}) = \frac{1 - cos x}{2}$ $sin^2 (x/2) = (1 - cos x)/2$ -->
$sin (\frac{x}{2}) = \pm \frac{\sqrt{(1 - cos x)}}{\sqrt{2}}$ $sin (x/2) = +- sqrt((1 - cos x))/(sqrt2)$ .

${cos}^{2} (\frac{x}{2}) = \frac{1 + cos x}{2}$ $cos^2 (x/2) = (1 + cos x)/2$ .
$cos (\frac{x}{2}) = \pm \sqrt{\frac{1 + cos x}{\sqrt{2}}}$ $cos (x/2) = +- sqrt((1 + cos x)/(sqrt2)$ .
$tan (\frac{x}{2}) = \sqrt{\frac{1 - cos x}{1 + cos x}}$ $tan (x/2) = sqrt((1 - cos x)/(1 + cos x))$ .

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Question #16d46

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