How do you verify the identity: $1 - cos 2x = tan x sin 2x$ ?

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Answer 1 · 2018-04-16T04:42:28

Here's how I proved it:

$1-cos2x =tanxsin2x$

I'll prove using the right hand side of the equation.

From the double angle identities, $sin2x=2sinxcosx$ :
$quadquadquadquadquadquadquadquadquad=sinx/cosx * 2sinxcosx$

Combine by multiplying:
$quadquadquadquadquadquadquadquadquad=(2sin^2xcancel(cosx))/cancel(cosx)$

$quadquadquadquadquadquadquadquadquad=2sin^2x$

From the Pythagorean Identities, $sin^2x = 1-cos^2x$ :
$quadquadquadquadquadquadquadquadquad=2(1-cos^2x)$

Simplify:
$quadquadquadquadquadquadquadquadquad=2-2cos^2x$

Factor out a $-1$ :
$quadquadquadquadquadquadquadquadquad=-1(-2+2cos^2x)$

Since we need $2cos^2x-1$ to get $cos2x$ , let's rewrite it so that we can get that:
$quadquadquadquadquadquadquadquadquad=-1(-1+2cos^2x-1)$

From the double angle identities, $cos^2x-1 = cos2x$ :
$quadquadquadquadquadquadquadquadquad=-1(-1+cos2x)$

Finally:
$1-cos2x=1-cos2x$

We have proved that $1-cos2x =tanxsin2x$ .

Hope this helps!

How do you verify the identity: 1 - cos 2x = tan x sin 2x1 - cos 2x = tan x sin 2x?