Trigonometry Examples

$\ln (\cos^{2} (x)) + \ln (1 + \tan^{2} (x)) = 0$

Step 1

Simplify the left side.

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Step 1.1

Factor $1$ out of $\ln (\cos^{2} (x)) + \ln (1 + \tan^{2} (x))$ .

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Step 1.1.1

Multiply by $1$ .

$1 \ln (\cos^{2} (x)) + \ln (1 + \tan^{2} (x)) = 0$

Step 1.1.2

Multiply by $1$ .

$1 \ln (\cos^{2} (x)) + 1 \ln (1 + \tan^{2} (x)) = 0$

Step 1.1.3

Factor $1$ out of $1 \ln (\cos^{2} (x)) + 1 \ln (1 + \tan^{2} (x))$ .

$1 (\ln (\cos^{2} (x)) + \ln (1 + \tan^{2} (x))) = 0$

Step 1.2

Rearrange terms.

$1 (\ln (\cos^{2} (x)) + \ln (\tan^{2} (x) + 1)) = 0$

Step 1.3

Apply pythagorean identity.

$1 (\ln (\cos^{2} (x)) + \ln (\sec^{2} (x))) = 0$

Step 1.4

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

$\ln (\cos^{2} (x)) + \ln (\sec^{2} (x)) = 0$

Step 1.5

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

$\ln (\cos^{2} (x) \sec^{2} (x)) = 0$

Step 1.6

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

$\ln (\cos^{2} (x) {(\frac{1}{\cos (x)})}^{2}) = 0$

Step 1.7

Apply the product rule to $\frac{1}{\cos (x)}$ .

$\ln (\cos^{2} (x) (\frac{1^{2}}{\cos^{2} (x)})) = 0$

Step 1.8

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

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Step 1.8.1

Cancel the common factor.

$\ln (\cos^{2} (x) (\frac{1^{2}}{\cos^{2} (x)})) = 0$

Step 1.8.2

Rewrite the expression.

$\ln (1^{2}) = 0$

Step 1.9

One to any power is one.

$\ln (1) = 0$

Step 1.10

The natural logarithm of $1$ is $0$ .

$0 = 0$

Step 2

The equation is always true for any value of $x$ .

All real numbers

Step 3

The result can be shown in multiple forms.

All real numbers

Interval Notation:

$(- \infty, \infty)$