Trigonometry Examples

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

Step 1

Replace the $\sin^{2} (x)$ with $1 - \cos^{2} (x)$ based on the $\sin^{2} (x) + \cos^{2} (x) = 1$ identity.

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

Step 2

Reorder the polynomial.

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

Step 3

Simplify the left side.

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Step 3.1

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

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Step 3.1.1

Simplify the expression.

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Step 3.1.1.1

Move $1$ .

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

Step 3.1.1.2

Reorder $- \cos^{2} (x)$ and $1$ .

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

Step 3.1.2

Apply pythagorean identity.

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

Step 3.1.3

Simplify each term.

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Step 3.1.3.1

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

$\sin^{2} (x) + {(\frac{\sin (x)}{\cos (x)})}^{2} = \sec^{2} (x)$

Step 3.1.3.2

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

$\sin^{2} (x) + \frac{\sin^{2} (x)}{\cos^{2} (x)} = \sec^{2} (x)$

Step 3.1.4

Convert from $\frac{\sin^{2} (x)}{\cos^{2} (x)}$ to $\tan^{2} (x)$ .

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

Step 4

Replace the $\sin^{2} (x)$ with $1 - \cos^{2} (x)$ based on the $\sin^{2} (x) + \cos^{2} (x) = 1$ identity.

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

Step 5

Reorder the polynomial.

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

Step 6

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

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Step 6.1

Simplify the expression.

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Step 6.1.1

Move $1$ .

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

Step 6.1.2

Reorder $- \cos^{2} (x)$ and $1$ .

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

Step 6.2

Apply pythagorean identity.

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

Step 6.3

Simplify each term.

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Step 6.3.1

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

$\sin^{2} (x) + {(\frac{\sin (x)}{\cos (x)})}^{2} = \sec^{2} (x)$

Step 6.3.2

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

$\sin^{2} (x) + \frac{\sin^{2} (x)}{\cos^{2} (x)} = \sec^{2} (x)$

Step 6.4

Convert from $\frac{\sin^{2} (x)}{\cos^{2} (x)}$ to $\tan^{2} (x)$ .

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

Step 7

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

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Step 7.1

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

$\sin^{2} (x) + \tan^{2} (x) - \sec^{2} (x) = 0$

Step 7.2

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

$\sin^{2} (x) - \sec^{2} (x) + \tan^{2} (x) = 0$

Step 7.3

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

$\sin^{2} (x) - (\sec^{2} (x)) + \tan^{2} (x) = 0$

Step 7.4

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

$\sin^{2} (x) - (\sec^{2} (x)) - 1 (- \tan^{2} (x)) = 0$

Step 7.5

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

$\sin^{2} (x) - (\sec^{2} (x) - \tan^{2} (x)) = 0$

Step 7.6

Apply pythagorean identity.

$\sin^{2} (x) - 1 \cdot 1 = 0$

Step 7.7

Multiply $- 1$ by $1$ .

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

Step 7.8

Reorder $\sin^{2} (x)$ and $- 1$ .

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

Step 7.9

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

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

Step 7.10

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

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

Step 7.11

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

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

Step 7.12

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

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

Step 7.13

Apply pythagorean identity.

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

Step 8

Divide each term in $- \cos^{2} (x) = 0$ by $- 1$ and simplify.

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Step 8.1

Divide each term in $- \cos^{2} (x) = 0$ by $- 1$ .

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

Step 8.2

Simplify the left side.

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Step 8.2.1

Dividing two negative values results in a positive value.

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

Step 8.2.2

Divide $\cos^{2} (x)$ by $1$ .

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

Step 8.3

Simplify the right side.

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Step 8.3.1

Divide $0$ by $- 1$ .

$\cos^{2} (x) = 0$

Step 9

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$\cos (x) = \pm \sqrt{0}$

Step 10

Simplify $\pm \sqrt{0}$ .

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Step 10.1

Rewrite $0$ as $0^{2}$ .

$\cos (x) = \pm \sqrt{0^{2}}$

Step 10.2

Pull terms out from under the radical, assuming positive real numbers.

$\cos (x) = \pm 0$

Step 10.3

Plus or minus $0$ is $0$ .

$\cos (x) = 0$

Step 11

Take the inverse cosine of both sides of the equation to extract $x$ from inside the cosine.

$x = \arccos (0)$

Step 12

Simplify the right side.

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Step 12.1

The exact value of $\arccos (0)$ is $\frac{π}{2}$ .

$x = \frac{π}{2}$

Step 13

The cosine function is positive in the first and fourth quadrants. To find the second solution, subtract the reference angle from $2 π$ to find the solution in the fourth quadrant.

$x = 2 π - \frac{π}{2}$

Step 14

Simplify $2 π - \frac{π}{2}$ .

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Step 14.1

To write $2 π$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$x = 2 π \cdot \frac{2}{2} - \frac{π}{2}$

Step 14.2

Combine fractions.

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Step 14.2.1

Combine $2 π$ and $\frac{2}{2}$ .

$x = \frac{2 π \cdot 2}{2} - \frac{π}{2}$

Step 14.2.2

Combine the numerators over the common denominator.

$x = \frac{2 π \cdot 2 - π}{2}$

Step 14.3

Simplify the numerator.

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Step 14.3.1

Multiply $2$ by $2$ .

$x = \frac{4 π - π}{2}$

Step 14.3.2

Subtract $π$ from $4 π$ .

$x = \frac{3 π}{2}$

Step 15

Find the period of $\cos (x)$ .

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Step 15.1

The period of the function can be calculated using $\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Step 15.2

Replace $b$ with $1$ in the formula for period.

$\frac{2 π}{| 1 |}$

Step 15.3

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{2 π}{1}$

Step 15.4

Divide $2 π$ by $1$ .

$2 π$

Step 16

The period of the $\cos (x)$ function is $2 π$ so values will repeat every $2 π$ radians in both directions.

$x = \frac{π}{2} + 2 π n, \frac{3 π}{2} + 2 π n$ , for any integer $n$

Step 17

Consolidate the answers.

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