How do you verify $(1-cos2x)/ tanx = sin2x$ ?

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Answer 1 · 2015-08-06T13:45:14

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

$cos2x = cos^2 x-sin^2 x = 1 - 2sin^2 x$
$sin 2x = 2sin x cos x$
$tan x = \frac{sin x}{cos x}$

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

Answer 2 · 2015-08-07T16:12:37

Starting with $(1-cos2x)/(tanx) = sin2x$ and using the identity $(1-cos2x)/2 = sin^2x$ :

$= (2sin^2x)/(tanx) = sin2x$

Of course, $tanx = (sinx)/(cosx)$ :

$= (2cosxsin^cancel(2)x)/(cancel(sinx)) = sin2x$

and:

$2sinxcosx = sinxcosx + cosxsinx$
$= sin(x+x) = sin2x$ :

$=> color(blue)(sin2x = sin2x)$

How do you verify (1-cos2x)/ tanx = sin2x(1-cos2x)/ tanx = sin2x?