Trigonometry Examples

$9 \sec^{2} (x) - 5 \tan^{2} (x) = 5 + 4 \sec^{2} (x)$

Step 1

Replace the $9 \sec^{2} (x)$ with $9 (1 + \tan^{2} (x))$ based on the $\tan^{2} (x) + 1 = \sec^{2} (x)$ identity.

$9 (1 + \tan^{2} (x)) - 5 \tan^{2} (x) = 5 + 4 \sec^{2} (x)$

Step 2

Simplify each term.

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Step 2.1

Apply the distributive property.

$9 \cdot 1 + 9 \tan^{2} (x) - 5 \tan^{2} (x) = 5 + 4 \sec^{2} (x)$

Step 2.2

Multiply $9$ by $1$ .

$9 + 9 \tan^{2} (x) - 5 \tan^{2} (x) = 5 + 4 \sec^{2} (x)$

Step 3

Subtract $5 \tan^{2} (x)$ from $9 \tan^{2} (x)$ .

$9 + 4 \tan^{2} (x) = 5 + 4 \sec^{2} (x)$

Step 4

Reorder the polynomial.

$4 \tan^{2} (x) + 9 = 5 + 4 \sec^{2} (x)$

Step 5

Move all terms containing $x$ to the left side of the equation.

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Step 5.1

Subtract $4 \sec^{2} (x)$ from both sides of the equation.

$4 \tan^{2} (x) + 9 - 4 \sec^{2} (x) = 5$

Step 5.2

Move $- 4 \sec^{2} (x)$ .

$4 \tan^{2} (x) - 4 \sec^{2} (x) + 9 = 5$

Step 5.3

Reorder $4 \tan^{2} (x)$ and $- 4 \sec^{2} (x)$ .

$- 4 \sec^{2} (x) + 4 \tan^{2} (x) + 9 = 5$

Step 5.4

Factor $- 4$ out of $- 4 \sec^{2} (x)$ .

$- 4 (\sec^{2} (x)) + 4 \tan^{2} (x) + 9 = 5$

Step 5.5

Factor $- 4$ out of $4 \tan^{2} (x)$ .

$- 4 (\sec^{2} (x)) - 4 (- \tan^{2} (x)) + 9 = 5$

Step 5.6

Factor $- 4$ out of $- 4 (\sec^{2} (x)) - 4 (- \tan^{2} (x))$ .

$- 4 (\sec^{2} (x) - \tan^{2} (x)) + 9 = 5$

Step 5.7

Apply pythagorean identity.

$- 4 \cdot 1 + 9 = 5$

Step 5.8

Multiply $- 4$ by $1$ .

$- 4 + 9 = 5$

Step 5.9

Add $- 4$ and $9$ .

$5 = 5$

Step 6

Since $5 = 5$ , 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:

$(- \infty, \infty)$