How can tan 4x be simplified or sec 2x?

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Answer 1 · 2015-08-07T16:28:08

Let's say we looked at $tan4x$ . We can use the following identities:

$tan4x = (sin4x)/(cos4x)$

$sin2x = 2sinxcosx$
$cos2x = cos^2x - sin^2x$

$=> (2sin2xcos2x)/(cos^2 2x - sin^2 2x)$

$= (4sinxcosx(cos^2x - sin^2x))/((cos^2x - sin^2x)^2 - (2sinxcosx)^2)$

$= (4sinxcosx(cos^2x - sin^2x))/((cos^2x - sin^2x)^2 - 4sin^2xcos^2x)$

$= color(blue)((4sinxcosx(1 - 2sin^2x))/((1 - 2sin^2x)^2 - 4sin^2xcos^2x))$

I don't know if you can get it any simpler; it's all $sinx$ and $cosx$ now, though.

You could also have used:
$tan(2x+2x) = (tan(2x)+tan(2x))/(1-tan(2x)tan(2x))$

$= (2tan(2x))/(1-tan^2(2x))$

but that's gonna be uglier to simplify (unless you stop here).

$sec(2x)$ is much simpler.

$= 1/(cos(2x)) = 1/(cos^2x - sin^2x) = color(blue)(1/(1-2sin^2x))$