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Arucerious · Jun 21, 2015

Can anyone help prove these identities? Thanks

1. (Sin(2x)+1)/cos(2x) = (cos(x) + sin(x))/cos(x) - sin(x)

2. cos(4x) = 8cos^4x - 8cos^2x + 1

3. (1-cos(x))/sin(x) = tanx/2

BreathingMaths · Jun 21, 2015

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

BreathingMaths · Jun 21, 2015

BreathingMaths · Jun 21, 2015

3.
$\frac{1- \cos x}{\sin x} = \frac{2 \sin^2 \frac{x}{2}}{2 \sin \frac{x}{2} \cos \frac{x}{2}} = \tan \frac{x}{2}$

Beware that:
$1- \cos x = (\sin \frac{x}{2})^2 + (\cos \frac{x}{2})^2 - \left((\cos \frac{x}{2})^2 - (\sin \frac{x}{2})^2 \right ) = 2(\sin \frac{x}{2})^2$

and
$1- \cos 2x = 2 \sin^2 x \Rightarrow 1- \cos x = 2 \sin^2 \frac{x}{2} \quad ($double angle formula$) \\\\$

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Arucerious

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