Trigonometry Examples

$2 \sec (x) \cos^{2} (x) - \csc^{2} (x) + \cot^{2} (x) = 0$

Step 1

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

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

Step 2

Reorder the polynomial.

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

Step 3

Simplify the left side.

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Step 3.1

Simplify $- \sin^{2} (x) - \csc^{2} (x) + \cot^{2} (x) + 1$ .

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Step 3.1.1

Simplify the expression.

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Step 3.1.1.1

Move $1$ .

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

Step 3.1.1.2

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

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

Step 3.1.2

Apply pythagorean identity.

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

Step 3.1.3

Simplify with factoring out.

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Step 3.1.3.1

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

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

Step 3.1.3.2

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

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

Step 3.1.3.3

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

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

Step 3.1.4

Apply pythagorean identity.

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

Step 3.1.5

Simplify with factoring out.

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Step 3.1.5.1

Multiply $- 1$ by $1$ .

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

Step 3.1.5.2

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

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

Step 3.1.5.3

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

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

Step 3.1.5.4

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

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

Step 3.1.5.5

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

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

Step 3.1.5.6

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

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

Step 3.1.6

Apply pythagorean identity.

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

Step 4

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

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Step 4.1

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

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

Step 4.2

Simplify the left side.

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Step 4.2.1

Dividing two negative values results in a positive value.

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

Step 4.2.2

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

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

Step 4.3

Simplify the right side.

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Step 4.3.1

Divide $0$ by $- 1$ .

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

Step 5

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

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

Step 6

Simplify $\pm \sqrt{0}$ .

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Step 6.1

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

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

Step 6.2

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

$\sin (x) = \pm 0$

Step 6.3

Plus or minus $0$ is $0$ .

$\sin (x) = 0$

Step 7

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

$x = \arcsin (0)$

Step 8

Simplify the right side.

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Step 8.1

The exact value of $\arcsin (0)$ is $0$ .

$x = 0$

Step 9

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

$x = π - 0$

Step 10

Subtract $0$ from $π$ .

$x = π$

Step 11

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

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Step 11.1

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

$\frac{2 π}{| b |}$

Step 11.2

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

$\frac{2 π}{| 1 |}$

Step 11.3

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

$\frac{2 π}{1}$

Step 11.4

Divide $2 π$ by $1$ .

$2 π$

Step 12

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

$x = 2 π n, π + 2 π n$ , for any integer $n$

Step 13

Consolidate the answers.

$x = π n$ , for any integer $n$