Precalculus Examples

$- \csc (x) + \csc (x) \sec^{2} (x)$

Step 1

Simplify with factoring out.

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Step 1.1

Reorder $- \csc (x)$ and $\csc (x) \sec^{2} (x)$ .

$\csc (x) \sec^{2} (x) - \csc (x)$

Step 1.2

Factor $\csc (x)$ out of $\csc (x) \sec^{2} (x)$ .

$\csc (x) (\sec^{2} (x)) - \csc (x)$

Step 1.3

Factor $\csc (x)$ out of $- \csc (x)$ .

$\csc (x) (\sec^{2} (x)) + \csc (x) \cdot - 1$

Step 1.4

Factor $\csc (x)$ out of $\csc (x) (\sec^{2} (x)) + \csc (x) \cdot - 1$ .

$\csc (x) (\sec^{2} (x) - 1)$

Step 2

Apply pythagorean identity.

$\csc (x) \tan^{2} (x)$

Step 3

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

$\frac{1}{\sin (x)} \tan^{2} (x)$

Step 4

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

$\frac{1}{\sin (x)} {(\frac{\sin (x)}{\cos (x)})}^{2}$

Step 5

Apply the product rule to $\frac{\sin (x)}{\cos (x)}$ .

$\frac{1}{\sin (x)} \cdot \frac{\sin^{2} (x)}{\cos^{2} (x)}$

Step 6

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

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Step 6.1

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

$\frac{1}{\sin (x)} \cdot \frac{\sin (x) \sin (x)}{\cos^{2} (x)}$

Step 6.2

Cancel the common factor.

$\frac{1}{\sin (x)} \cdot \frac{\sin (x) \sin (x)}{\cos^{2} (x)}$

Step 6.3

Rewrite the expression.

$\frac{\sin (x)}{\cos^{2} (x)}$

Step 7

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

$\frac{\sin (x)}{\cos (x) \cos (x)}$

Step 8

Separate fractions.

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

Step 9

Convert from $\frac{\sin (x)}{\cos (x)}$ to $\tan (x)$ .

$\frac{1}{\cos (x)} \tan (x)$

Step 10

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

$\sec (x) \tan (x)$