Algebra Examples

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

Step 1

Write $f (x) = \frac{1 - \cos^{2} (4 x)}{\tan (4 x)} \cdot \sec (4 x)$ as an equation.

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

Step 2

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

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Step 2.1

Apply pythagorean identity.

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

Step 2.2

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

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

Step 2.3

Multiply the numerator by the reciprocal of the denominator.

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

Step 2.4

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

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Step 2.4.1

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

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

Step 2.4.2

Cancel the common factor.

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

Step 2.4.3

Rewrite the expression.

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

Step 2.5

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

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

Step 2.6

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

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Step 2.6.1

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

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

Step 2.6.2

Cancel the common factor.

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

Step 2.6.3

Rewrite the expression.

$y = \sin (4 x)$