Trigonometry Examples

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

Step 1

Simplify the left side.

Tap for more steps...

Step 1.1

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

Tap for more steps...

Step 1.1.1

Simplify the expression.

Tap for more steps...

Step 1.1.1.1

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

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

Step 1.1.1.2

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

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

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 $a = \sec^{2} (x)$ and $b = \tan^{2} (x)$ .

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

Step 1.1.3

Apply pythagorean identity.

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

Step 1.1.4

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

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

Step 2

Replace the $\sec^{2} (x)$ with $1 + \tan^{2} (x)$ based on the $\tan^{2} (x) + 1 = \sec^{2} (x)$ identity.

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

Step 3

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

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

Step 4

Reorder the polynomial.

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

Step 5

Move all terms containing $\tan (x)$ to the left side of the equation.

Tap for more steps...

Step 5.1

Subtract $2 \tan^{2} (x)$ from both sides of the equation.

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

Step 5.2

Combine the opposite terms in $2 \tan^{2} (x) + 1 - 2 \tan^{2} (x)$ .

Tap for more steps...

Step 5.2.1

Subtract $2 \tan^{2} (x)$ from $2 \tan^{2} (x)$ .

$0 + 1 = 1$

Step 5.2.2

Add $0$ and $1$ .

$1 = 1$

Step 6

Since $1 = 1$ , the equation will always be true for any value of $x$ .

All real numbers

Step 7

The result can be shown in multiple forms.

All real numbers

Interval Notation:

$(- \infty, \infty)$