Trigonometry Examples

$\frac{9 + \sin^{2} (x)}{\cos^{2} (x)} = 10 \sec^{2} (x) - 1$

Step 1

Move all the expressions to the left side of the equation.

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Step 1.1

Subtract $10 \sec^{2} (x)$ from both sides of the equation.

$\frac{9 + \sin^{2} (x)}{\cos^{2} (x)} - 10 \sec^{2} (x) = - 1$

Step 1.2

Add $1$ to both sides of the equation.

$\frac{9 + \sin^{2} (x)}{\cos^{2} (x)} - 10 \sec^{2} (x) + 1 = 0$

Step 2

Simplify $\frac{9 + \sin^{2} (x)}{\cos^{2} (x)} - 10 \sec^{2} (x) + 1$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

Rewrite $\sec (x)$ in terms of sines and cosines.

$\frac{9 + \sin^{2} (x)}{\cos^{2} (x)} - 10 {(\frac{1}{\cos (x)})}^{2} + 1 = 0$

Step 2.1.2

Apply the product rule to $\frac{1}{\cos (x)}$ .

$\frac{9 + \sin^{2} (x)}{\cos^{2} (x)} - 10 \frac{1^{2}}{\cos^{2} (x)} + 1 = 0$

Step 2.1.3

One to any power is one.

$\frac{9 + \sin^{2} (x)}{\cos^{2} (x)} - 10 \frac{1}{\cos^{2} (x)} + 1 = 0$

Step 2.1.4

Combine $- 10$ and $\frac{1}{\cos^{2} (x)}$ .

$\frac{9 + \sin^{2} (x)}{\cos^{2} (x)} + \frac{- 10}{\cos^{2} (x)} + 1 = 0$

Step 2.1.5

Move the negative in front of the fraction.

$\frac{9 + \sin^{2} (x)}{\cos^{2} (x)} - \frac{10}{\cos^{2} (x)} + 1 = 0$

Step 2.2

Combine the numerators over the common denominator.

$1 + \frac{9 + \sin^{2} (x) - 10}{\cos^{2} (x)} = 0$

Step 2.3

Subtract $10$ from $9$ .

$1 + \frac{- 1 + \sin^{2} (x)}{\cos^{2} (x)} = 0$

Step 2.4

Rewrite $- 1$ as $- 1 (1)$ .

$1 + \frac{- 1 (1) + \sin^{2} (x)}{\cos^{2} (x)} = 0$

Step 2.5

Factor $- 1$ out of $\sin^{2} (x)$ .

$1 + \frac{- 1 (1) - 1 (- \sin^{2} (x))}{\cos^{2} (x)} = 0$

Step 2.6

Factor $- 1$ out of $- 1 (1) - 1 (- \sin^{2} (x))$ .

$1 + \frac{- 1 (1 - \sin^{2} (x))}{\cos^{2} (x)} = 0$

Step 2.7

Rewrite $- 1 (1 - \sin^{2} (x))$ as $- (1 - \sin^{2} (x))$ .

$1 + \frac{- (1 - \sin^{2} (x))}{\cos^{2} (x)} = 0$

Step 2.8

Apply pythagorean identity.

$1 + \frac{- \cos^{2} (x)}{\cos^{2} (x)} = 0$

Step 2.9

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

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Step 2.9.1

Cancel the common factor.

$1 + \frac{- \cos^{2} (x)}{\cos^{2} (x)} = 0$

Step 2.9.2

Divide $- 1$ by $1$ .

$1 - 1 = 0$

Step 2.10

Subtract $1$ from $1$ .

$0 = 0$

Step 3

Since $0 = 0$ , the equation will always be true for any value of $x$ .

All real numbers

Step 4

The result can be shown in multiple forms.

All real numbers

Interval Notation:

$(- \infty, \infty)$