Precalculus Examples

$\sec (x) - \tan (x) \sin (x) + \frac{\sin (2 x)}{2 \sin (x)}$

Step 1

Simplify terms.

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Step 1.1

Simplify each term.

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Step 1.1.1

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

$\frac{1}{\cos (x)} - \tan (x) \sin (x) + \frac{\sin (2 x)}{2 \sin (x)}$

Step 1.1.2

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

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

Step 1.1.3

Multiply $- \frac{\sin (x)}{\cos (x)} \sin (x)$ .

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Step 1.1.3.1

Combine $\sin (x)$ and $\frac{\sin (x)}{\cos (x)}$ .

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

Step 1.1.3.2

Raise $\sin (x)$ to the power of $1$ .

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

Step 1.1.3.3

Raise $\sin (x)$ to the power of $1$ .

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

Step 1.1.3.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 1.1.3.5

Add $1$ and $1$ .

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

Step 1.1.4

Apply the sine double-angle identity.

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

Step 1.1.5

Cancel the common factor of $2$ .

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Step 1.1.5.1

Cancel the common factor.

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

Step 1.1.5.2

Rewrite the expression.

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

Step 1.1.6

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

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Step 1.1.6.1

Cancel the common factor.

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

Step 1.1.6.2

Divide $\cos (x)$ by $1$ .

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

Step 1.2

Combine the numerators over the common denominator.

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

Step 2

Apply pythagorean identity.

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

Step 3

Simplify terms.

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Step 3.1

Cancel the common factor of $\cos^{2} (x)$ and $\cos (x)$ .

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Step 3.1.1

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

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

Step 3.1.2

Cancel the common factors.

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Step 3.1.2.1

Multiply by $1$ .

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

Step 3.1.2.2

Cancel the common factor.

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

Step 3.1.2.3

Rewrite the expression.

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

Step 3.1.2.4

Divide $\cos (x)$ by $1$ .

$\cos (x) + \cos (x)$

Step 3.2

Add $\cos (x)$ and $\cos (x)$ .

$2 \cos (x)$