Simplify f(x)=(1-cos(4x)^2)/(tan(4x))*sec(4x)

$f (x) = \frac{1 - \cos^{2} (4 x)}{\tan (4 x)} \cdot \sec (4 x)$

Apply pythagorean identity.

$f (x) = \frac{\sin^{2} (4 x)}{\tan (4 x)} \cdot \sec (4 x)$

Rewrite $\tan (4 x)$ in terms of sines and cosines.

$f (x) = \frac{\sin^{2} (4 x)}{\frac{\sin (4 x)}{\cos (4 x)}} \cdot \sec (4 x)$

Multiply the numerator by the reciprocal of the denominator.

$f (x) = \sin^{2} (4 x) (\frac{\cos (4 x)}{\sin (4 x)}) \cdot \sec (4 x)$

Cancel the common factor of $\sin (4 x)$ .

Factor $\sin (4 x)$ out of $\sin^{2} (4 x)$ .

$f (x) = \sin (4 x) \sin (4 x) (\frac{\cos (4 x)}{\sin (4 x)}) \cdot \sec (4 x)$

Cancel the common factor.

$f (x) = \sin (4 x) \sin (4 x) (\frac{\cos (4 x)}{\sin (4 x)}) \cdot \sec (4 x)$

Rewrite the expression.

$f (x) = \sin (4 x) \cos (4 x) \cdot \sec (4 x)$

Rewrite $\sec (4 x)$ in terms of sines and cosines.

$f (x) = \sin (4 x) \cos (4 x) \cdot \frac{1}{\cos (4 x)}$

Cancel the common factor of $\cos (4 x)$ .

Factor $\cos (4 x)$ out of $\sin (4 x) \cos (4 x)$ .

$f (x) = \cos (4 x) \sin (4 x) \cdot \frac{1}{\cos (4 x)}$

Cancel the common factor.

$f (x) = \cos (4 x) \sin (4 x) \cdot \frac{1}{\cos (4 x)}$

Rewrite the expression.

$f (x) = \sin (4 x)$