Precalculus Examples

$\ln (\cos^{2} (x)) + \ln (1 + \tan (x^{2}))$

Step 1

Use the product property of logarithms, $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

$\ln (\cos^{2} (x) (1 + \tan (x^{2})))$

Step 2

Rewrite $\tan (x^{2})$ in terms of sines and cosines.

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

Step 3

Simplify terms.

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Step 3.1

Apply the distributive property.

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

Step 3.2

Multiply $\cos^{2} (x)$ by $1$ .

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

Step 3.3

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

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

Step 4

Simplify each term.

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Step 4.1

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

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

Step 4.2

Separate fractions.

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

Step 4.3

Rewrite $\frac{\cos (x)}{\cos (x^{2})}$ as a product.

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

Step 4.4

Write $\cos (x)$ as a fraction with denominator $1$ .

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

Step 4.5

Simplify.

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Step 4.5.1

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

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

Step 4.5.2

Convert from $\frac{1}{\cos (x^{2})}$ to $\sec (x^{2})$ .

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

Step 4.6

Divide $\cos (x) \sin (x^{2})$ by $1$ .

$\ln (\cos^{2} (x) + \cos (x) \sin (x^{2}) (\cos (x) \sec (x^{2})))$

Step 4.7

Multiply $\cos (x) \sin (x^{2}) (\cos (x) \sec (x^{2}))$ .

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Step 4.7.1

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

$\ln (\cos^{2} (x) + \cos^{1} (x) \cos (x) \sin (x^{2}) \sec (x^{2}))$

Step 4.7.2

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

$\ln (\cos^{2} (x) + \cos^{1} (x) \cos^{1} (x) \sin (x^{2}) \sec (x^{2}))$

Step 4.7.3

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

$\ln (\cos^{2} (x) + {\cos (x)}^{1 + 1} \sin (x^{2}) \sec (x^{2}))$

Step 4.7.4

Add $1$ and $1$ .

$\ln (\cos^{2} (x) + \cos^{2} (x) \sin (x^{2}) \sec (x^{2}))$