Trigonometry Examples

$\frac{7 - 7 \tan^{4} (x)}{\sec^{2} (x)} = 7 (1 - \tan^{2} (x))$

Step 1

Simplify $\frac{7 - 7 \tan^{4} (x)}{\sec^{2} (x)}$ .

Tap for more steps...

Step 1.1

Simplify the numerator.

Tap for more steps...

Step 1.1.1

Factor $7$ out of $7 - 7 \tan^{4} (x)$ .

Tap for more steps...

Step 1.1.1.1

Factor $7$ out of $7$ .

$\frac{7 (1) - 7 \tan^{4} (x)}{\sec^{2} (x)} = 7 (1 - \tan^{2} (x))$

Step 1.1.1.2

Factor $7$ out of $- 7 \tan^{4} (x)$ .

$\frac{7 (1) + 7 (- \tan^{4} (x))}{\sec^{2} (x)} = 7 (1 - \tan^{2} (x))$

Step 1.1.1.3

Factor $7$ out of $7 (1) + 7 (- \tan^{4} (x))$ .

$\frac{7 (1 - \tan^{4} (x))}{\sec^{2} (x)} = 7 (1 - \tan^{2} (x))$

Step 1.1.2

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

$\frac{7 (1^{2} - \tan^{4} (x))}{\sec^{2} (x)} = 7 (1 - \tan^{2} (x))$

Step 1.1.3

Rewrite $\tan^{4} (x)$ as ${(\tan^{2} (x))}^{2}$ .

$\frac{7 (1^{2} - {(\tan^{2} (x))}^{2})}{\sec^{2} (x)} = 7 (1 - \tan^{2} (x))$

Step 1.1.4

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

$\frac{7 ((1 + \tan^{2} (x)) (1 - \tan^{2} (x)))}{\sec^{2} (x)} = 7 (1 - \tan^{2} (x))$

Step 1.1.5

Simplify.

Tap for more steps...

Step 1.1.5.1

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

$\frac{7 ((1 + \tan^{2} (x)) (1^{2} - \tan^{2} (x)))}{\sec^{2} (x)} = 7 (1 - \tan^{2} (x))$

Step 1.1.5.2

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

$\frac{7 ((1 + \tan^{2} (x)) (1 + \tan (x)) (1 - \tan (x)))}{\sec^{2} (x)} = 7 (1 - \tan^{2} (x))$

$\frac{7 (1 + \tan^{2} (x)) (1 + \tan (x)) (1 - \tan (x))}{\sec^{2} (x)} = 7 (1 - \tan^{2} (x))$

Step 1.1.6

Rearrange terms.

$\frac{7 (\tan^{2} (x) + 1) (1 + \tan (x)) (1 - \tan (x))}{\sec^{2} (x)} = 7 (1 - \tan^{2} (x))$

Step 1.1.7

Apply pythagorean identity.

$\frac{7 \sec^{2} (x) (1 + \tan (x)) (1 - \tan (x))}{\sec^{2} (x)} = 7 (1 - \tan^{2} (x))$

Step 1.2

Simplify terms.

Tap for more steps...

Step 1.2.1

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

Tap for more steps...

Step 1.2.1.1

Cancel the common factor.

$\frac{7 \sec^{2} (x) (1 + \tan (x)) (1 - \tan (x))}{\sec^{2} (x)} = 7 (1 - \tan^{2} (x))$

Step 1.2.1.2

Divide $(7 (1 + \tan (x))) (1 - \tan (x))$ by $1$ .

$(7 (1 + \tan (x))) (1 - \tan (x)) = 7 (1 - \tan^{2} (x))$

Step 1.2.2

Apply the distributive property.

$(7 \cdot 1 + 7 \tan (x)) (1 - \tan (x)) = 7 (1 - \tan^{2} (x))$

Step 1.2.3

Multiply $7$ by $1$ .

$(7 + 7 \tan (x)) (1 - \tan (x)) = 7 (1 - \tan^{2} (x))$

Step 1.3

Expand $(7 + 7 \tan (x)) (1 - \tan (x))$ using the FOIL Method.

Tap for more steps...

Step 1.3.1

Apply the distributive property.

$7 (1 - \tan (x)) + 7 \tan (x) (1 - \tan (x)) = 7 (1 - \tan^{2} (x))$

Step 1.3.2

Apply the distributive property.

$7 \cdot 1 + 7 (- \tan (x)) + 7 \tan (x) (1 - \tan (x)) = 7 (1 - \tan^{2} (x))$

Step 1.3.3

Apply the distributive property.

$7 \cdot 1 + 7 (- \tan (x)) + 7 \tan (x) \cdot 1 + 7 \tan (x) (- \tan (x)) = 7 (1 - \tan^{2} (x))$

Step 1.4

Simplify terms.

Tap for more steps...

Step 1.4.1

Combine the opposite terms in $7 \cdot 1 + 7 (- \tan (x)) + 7 \tan (x) \cdot 1 + 7 \tan (x) (- \tan (x))$ .

Tap for more steps...

Step 1.4.1.1

Reorder the factors in the terms $7 (- \tan (x))$ and $7 \tan (x) \cdot 1$ .

$7 \cdot 1 - 1 \cdot 7 \tan (x) + 1 \cdot 7 \tan (x) + 7 \tan (x) (- \tan (x)) = 7 (1 - \tan^{2} (x))$

Step 1.4.1.2

Add $- 1 \cdot 7 \tan (x)$ and $1 \cdot 7 \tan (x)$ .

$7 \cdot 1 + 0 + 7 \tan (x) (- \tan (x)) = 7 (1 - \tan^{2} (x))$

Step 1.4.1.3

Add $7 \cdot 1$ and $0$ .

$7 \cdot 1 + 7 \tan (x) (- \tan (x)) = 7 (1 - \tan^{2} (x))$

Step 1.4.2

Simplify each term.

Tap for more steps...

Step 1.4.2.1

Multiply $7$ by $1$ .

$7 + 7 \tan (x) (- \tan (x)) = 7 (1 - \tan^{2} (x))$

Step 1.4.2.2

Multiply $7 \tan (x) (- \tan (x))$ .

Tap for more steps...

Step 1.4.2.2.1

Multiply $- 1$ by $7$ .

$7 - 7 \tan (x) \tan (x) = 7 (1 - \tan^{2} (x))$

Step 1.4.2.2.2

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

$7 - 7 (\tan^{1} (x) \tan (x)) = 7 (1 - \tan^{2} (x))$

Step 1.4.2.2.3

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

$7 - 7 (\tan^{1} (x) \tan^{1} (x)) = 7 (1 - \tan^{2} (x))$

Step 1.4.2.2.4

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

$7 - 7 {\tan (x)}^{1 + 1} = 7 (1 - \tan^{2} (x))$

Step 1.4.2.2.5

Add $1$ and $1$ .

$7 - 7 \tan^{2} (x) = 7 (1 - \tan^{2} (x))$

Step 2

Simplify $7 (1 - \tan^{2} (x))$ .

Tap for more steps...

Step 2.1

Apply the distributive property.

$7 - 7 \tan^{2} (x) = 7 \cdot 1 + 7 (- \tan^{2} (x))$

Step 2.2

Multiply.

Tap for more steps...

Step 2.2.1

Multiply $7$ by $1$ .

$7 - 7 \tan^{2} (x) = 7 + 7 (- \tan^{2} (x))$

Step 2.2.2

Multiply $- 1$ by $7$ .

$7 - 7 \tan^{2} (x) = 7 - 7 \tan^{2} (x)$

Step 3

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

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

Step 4