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Step-by-step Solution

Simplify $\frac{1}{\sin\left(2x\right)^2\cos\left(2x\right)^4}$

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Answer

$\csc\left(x\right)^2\frac{1}{\left(1-2\sin\left(x\right)^2\right)^4}\cdot\frac{\frac{1}{4}}{\cos\left(x\right)^2}$

Step-by-step explanation

Problem to solve:

$\left(1/\left(sin\left(2\cdot x\right)^2\cdot cos\left(2\cdot x\right)^4\right)\right)$
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Using the sine double-angle identity

$\frac{1}{\left(2\sin\left(x\right)\cos\left(x\right)\right)^2\cos\left(2x\right)^4}$
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Applying an identity of double-angle cosine

$\frac{1}{\left(2\sin\left(x\right)\cos\left(x\right)\right)^2\left(1-2\sin\left(x\right)^2\right)^4}$

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Answer

$\csc\left(x\right)^2\frac{1}{\left(1-2\sin\left(x\right)^2\right)^4}\cdot\frac{\frac{1}{4}}{\cos\left(x\right)^2}$
$\left(1/\left(sin\left(2\cdot x\right)^2\cdot cos\left(2\cdot x\right)^4\right)\right)$

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Time to solve it:

~ 0.66 seconds

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