Trigonometry Examples

$1 - \tan^{2} (x) = \frac{\cos (2 x)}{\cos^{2} (x)}$

Step 1

Simplify $\frac{\cos (2 x)}{\cos^{2} (x)}$ .

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Step 1.1

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

$1 - \tan^{2} (x) = \frac{\cos (2 x)}{\cos (x) \cos (x)}$

Step 1.2

Separate fractions.

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

Step 1.3

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

$1 - \tan^{2} (x) = \frac{1}{\cos (x)} (\cos (2 x) \frac{1}{\cos (x)})$

Step 1.4

Write $\cos (2 x)$ as a fraction with denominator $1$ .

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

Step 1.5

Simplify.

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Step 1.5.1

Divide $\cos (2 x)$ by $1$ .

$1 - \tan^{2} (x) = \frac{1}{\cos (x)} (\cos (2 x) \frac{1}{\cos (x)})$

Step 1.5.2

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

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

Step 1.6

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

$1 - \tan^{2} (x) = \sec (x) (\cos (2 x) \sec (x))$

Step 1.7

Multiply $\sec (x) (\cos (2 x) \sec (x))$ .

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Step 1.7.1

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

$1 - \tan^{2} (x) = \sec^{1} (x) \sec (x) \cos (2 x)$

Step 1.7.2

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

$1 - \tan^{2} (x) = \sec^{1} (x) \sec^{1} (x) \cos (2 x)$

Step 1.7.3

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

$1 - \tan^{2} (x) = {\sec (x)}^{1 + 1} \cos (2 x)$

Step 1.7.4

Add $1$ and $1$ .

$1 - \tan^{2} (x) = \sec^{2} (x) \cos (2 x)$

Step 2

Graph each side of the equation. The solution is the x-value of the point of intersection.

$x \approx 0, - 0.1, 0.1, \frac{1}{5}, - 0.3, 0.3, - \frac{2}{5}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$x \approx 0, - 0.1, 0.1, \frac{1}{5}, - 0.3, 0.3, - \frac{2}{5}$

Decimal Form:

$x \approx 0, - 0.1, 0.1, 0.2, - 0.3, 0.3, - 0.4$

Step 4