Verify these identities are true.

$\displaystyle cos(4\theta) = cos^4\theta - 6 sin^2\theta cos^2\theta+sin^4\theta$

$\displaystyle cos^4x=\frac{1}{8}(3 + 4cos(2x) + cos(4x))$

$\displaystyle sec^2\frac{U}{2} = \frac{2sec U}{secU+1}$

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For the first one I am thinking, turn all the sin's into cos'. And try to get it look like $\displaystyle cos4\theta = 2cos^2\theta - 1$

So I converted the all the sin^4 to sin^2 * sin^2, replacing it with 1-cos^2 each, along with the one in the original to get an equation that looks like this:

$\displaystyle cos4\theta = cos^4\theta-6(1-cos^2\theta)cos^2\theta+(1-cos^2\theta)(1-cos^2\theta)$

and tried to simplify...

$\displaystyle cos4\theta = cos^4\theta-(6cos^2\theta-6cos^4\theta)+(1-cos^2\theta)(1-cos^2\theta)$

however I keep getting lost and having difficulty.

The second one I'm completely lost on.

The third one is clearly a half angle identity but how to simplify it is beyond me at this moment.

My professor gave us 40 identities to study for the upcoming test and I've been working on them all night. I'm just a tiny bit frustrated at this point.