Trigonometry Examples

$\frac{\tan^{2} (x) - 1}{2} - \sec (x) + \sqrt{2} = 0$

Step 1

Simplify $\frac{\tan^{2} (x) - 1}{2} - \sec (x) + \sqrt{2}$ .

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Step 1.1

Simplify the numerator.

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Step 1.1.1

Rewrite $1$ as $1^{2}$ .

$\frac{\tan^{2} (x) - 1^{2}}{2} - \sec (x) + \sqrt{2} = 0$

Step 1.1.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = \tan (x)$ and $b = 1$ .

$\frac{(\tan (x) + 1) (\tan (x) - 1)}{2} - \sec (x) + \sqrt{2} = 0$

Step 1.2

To write $- \sec (x)$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{(\tan (x) + 1) (\tan (x) - 1)}{2} - \sec (x) \cdot \frac{2}{2} + \sqrt{2} = 0$

Step 1.3

Simplify terms.

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Step 1.3.1

Combine $- \sec (x)$ and $\frac{2}{2}$ .

$\frac{(\tan (x) + 1) (\tan (x) - 1)}{2} + \frac{- \sec (x) \cdot 2}{2} + \sqrt{2} = 0$

Step 1.3.2

Combine the numerators over the common denominator.

$\frac{(\tan (x) + 1) (\tan (x) - 1) - \sec (x) \cdot 2}{2} + \sqrt{2} = 0$

Step 1.4

Simplify the numerator.

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Step 1.4.1

Expand $(\tan (x) + 1) (\tan (x) - 1)$ using the FOIL Method.

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Step 1.4.1.1

Apply the distributive property.

$\frac{\tan (x) (\tan (x) - 1) + 1 (\tan (x) - 1) - \sec (x) \cdot 2}{2} + \sqrt{2} = 0$

Step 1.4.1.2

Apply the distributive property.

$\frac{\tan (x) \tan (x) + \tan (x) \cdot - 1 + 1 (\tan (x) - 1) - \sec (x) \cdot 2}{2} + \sqrt{2} = 0$

Step 1.4.1.3

Apply the distributive property.

$\frac{\tan (x) \tan (x) + \tan (x) \cdot - 1 + 1 \tan (x) + 1 \cdot - 1 - \sec (x) \cdot 2}{2} + \sqrt{2} = 0$

Step 1.4.2

Simplify and combine like terms.

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Step 1.4.2.1

Simplify each term.

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Step 1.4.2.1.1

Multiply $\tan (x) \tan (x)$ .

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Step 1.4.2.1.1.1

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

$\frac{\tan^{1} (x) \tan (x) + \tan (x) \cdot - 1 + 1 \tan (x) + 1 \cdot - 1 - \sec (x) \cdot 2}{2} + \sqrt{2} = 0$

Step 1.4.2.1.1.2

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

$\frac{\tan^{1} (x) \tan^{1} (x) + \tan (x) \cdot - 1 + 1 \tan (x) + 1 \cdot - 1 - \sec (x) \cdot 2}{2} + \sqrt{2} = 0$

Step 1.4.2.1.1.3

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

$\frac{{\tan (x)}^{1 + 1} + \tan (x) \cdot - 1 + 1 \tan (x) + 1 \cdot - 1 - \sec (x) \cdot 2}{2} + \sqrt{2} = 0$

Step 1.4.2.1.1.4

Add $1$ and $1$ .

$\frac{\tan^{2} (x) + \tan (x) \cdot - 1 + 1 \tan (x) + 1 \cdot - 1 - \sec (x) \cdot 2}{2} + \sqrt{2} = 0$

Step 1.4.2.1.2

Move $- 1$ to the left of $\tan (x)$ .

$\frac{\tan^{2} (x) - 1 \cdot \tan (x) + 1 \tan (x) + 1 \cdot - 1 - \sec (x) \cdot 2}{2} + \sqrt{2} = 0$

Step 1.4.2.1.3

Rewrite $- 1 \tan (x)$ as $- \tan (x)$ .

$\frac{\tan^{2} (x) - \tan (x) + 1 \tan (x) + 1 \cdot - 1 - \sec (x) \cdot 2}{2} + \sqrt{2} = 0$

Step 1.4.2.1.4

Multiply $\tan (x)$ by $1$ .

$\frac{\tan^{2} (x) - \tan (x) + \tan (x) + 1 \cdot - 1 - \sec (x) \cdot 2}{2} + \sqrt{2} = 0$

Step 1.4.2.1.5

Multiply $- 1$ by $1$ .

$\frac{\tan^{2} (x) - \tan (x) + \tan (x) - 1 - \sec (x) \cdot 2}{2} + \sqrt{2} = 0$

Step 1.4.2.2

Add $- \tan (x)$ and $\tan (x)$ .

$\frac{\tan^{2} (x) + 0 - 1 - \sec (x) \cdot 2}{2} + \sqrt{2} = 0$

Step 1.4.2.3

Add $\tan^{2} (x)$ and $0$ .

$\frac{\tan^{2} (x) - 1 - \sec (x) \cdot 2}{2} + \sqrt{2} = 0$

Step 1.4.3

Multiply $2$ by $- 1$ .

$\frac{\tan^{2} (x) - 1 - 2 \sec (x)}{2} + \sqrt{2} = 0$

Step 2

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

$x = \frac{π}{4} + 2 π n, \frac{7 π}{4} + 2 π n$ , for any integer $n$

Step 3