conversion of a trigonometic function

by frensel
Tags: conversion, function, trigonometic
 P: 20 1. The problem statement, all variables and given/known data How to convert $\tan(x)\sin(\frac{x}{2})+\cos(\frac{x}{2})$ to $\frac{\tan(x)}{\sqrt{2(1-\cos(x))}}$ 2. Relevant equations 3. The attempt at a solution I can convert it to this form: $\frac{\cos(\frac{x}{2})}{\cos(x)}$ $\tan(x)\sin(\frac{x}{2})+\cos(\frac{x}{2})$ =$\frac{\sin(x)}{\cos(x)}\sin(\frac{x}{2})+ \cos(\frac{x}{2})$ =$\frac{1}{\cos(x)}\left(\sin(x)\sin(\frac{x}{2})+ \cos(x)\cos(\frac{x}{2})\right)$ using angle sum and difference identities, we get $\left(\sin(x)\sin(\frac{x}{2})+ \cos(x)\cos(\frac{x}{2})\right) = \cos(x - \frac{x}{2}) = \cos(\frac{x}{2})$ therefore, we have $\tan(x)\sin(\frac{x}{2})+\cos(\frac{x}{2}) = \frac{\cos(\frac{x}{2})}{\cos(x)}$ 1. The problem statement, all variables and given/known data 2. Relevant equations 3. The attempt at a solution
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hi frensel
 Quote by frensel I can convert it to this form: $\frac{\cos(\frac{x}{2})}{\cos(x)}$
hint: multiply by sin(x)/sin(x), and use the half-angle identities
P: 20
 Quote by tiny-tim hi frensel hint: multiply by sin(x)/sin(x), and use the half-angle identities
I got it, thx!

$\frac{\cos(\frac{x}{2})}{\cos{x}}$
$= \frac{\sin(x)}{\sin(x)}\frac{\cos(\frac{x}{2})}{ \cos{x}}$
$=\tan(x)\frac{\cos(\frac{x}{2})}{\sin(x)}$

using double-angle formula, we have
$\tan(x)\frac{\cos(\frac{x}{2})}{\sin(x)}$
$=\tan(x)\frac{\cos(\frac{x}{2})}{2\sin(\frac{x}{2})\cos(\frac{x}{2})}$
$=\tan(x)\frac{1}{2\sin(\frac{x}{2})}$

finally, using half-angle formula (assuming $\sin(\frac{x}{2})>0$), then

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

Well, although I get the correct result, the calculation is so complicated. Is there any easier way to convert the above trigonometric function?

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conversion of a trigonometic function

you could work backwards (from the answer) …

(tan / 2sin1/2) - cos1/2 = … ?

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