- #1

frensel

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## Homework Statement

How to convert

[itex]\tan(x)\sin(\frac{x}{2})+\cos(\frac{x}{2})[/itex]

to

[itex]\frac{\tan(x)}{\sqrt{2(1-\cos(x))}}[/itex]

## Homework Equations

## The Attempt at a Solution

I can convert it to this form: [itex]\frac{\cos(\frac{x}{2})}{\cos(x)}[/itex]

[itex]\tan(x)\sin(\frac{x}{2})+\cos(\frac{x}{2})[/itex]

=[itex]\frac{\sin(x)}{\cos(x)}\sin(\frac{x}{2})+ \cos(\frac{x}{2})[/itex]

=[itex]\frac{1}{\cos(x)}\left(\sin(x)\sin(\frac{x}{2})+ \cos(x)\cos(\frac{x}{2})\right)[/itex]

using angle sum and difference identities, we get

[itex]\left(\sin(x)\sin(\frac{x}{2})+ \cos(x)\cos(\frac{x}{2})\right) = \cos(x - \frac{x}{2}) = \cos(\frac{x}{2})[/itex]

therefore, we have

[itex]\tan(x)\sin(\frac{x}{2})+\cos(\frac{x}{2}) = \frac{\cos(\frac{x}{2})}{\cos(x)}[/itex]