Trigonometry Examples

$\sin (x) \cos (x) = \cot (x)$

Step 1

Simplify the right side.

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Step 1.1

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

$\sin (x) \cos (x) = \frac{\cos (x)}{\sin (x)}$

Step 2

Multiply both sides of the equation by $\sin (x)$ .

$\sin (x) (\sin (x) \cos (x)) = \sin (x) \frac{\cos (x)}{\sin (x)}$

Step 3

Multiply $\sin (x) (\sin (x) \cos (x))$ .

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Step 3.1

Raise $\sin (x)$ to the power of $1$ .

$\sin^{1} (x) \sin (x) \cos (x) = \sin (x) \frac{\cos (x)}{\sin (x)}$

Step 3.2

Raise $\sin (x)$ to the power of $1$ .

$\sin^{1} (x) \sin^{1} (x) \cos (x) = \sin (x) \frac{\cos (x)}{\sin (x)}$

Step 3.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 3.4

Add $1$ and $1$ .

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

Step 4

Cancel the common factor of $\sin (x)$ .

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Step 4.1

Cancel the common factor.

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

Step 4.2

Rewrite the expression.

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

Step 5

Subtract $\cos (x)$ from both sides of the equation.

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

Step 6

Simplify $\sin^{2} (x) \cos (x) - \cos (x)$ .

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Step 6.1

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

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Step 6.1.1

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

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

Step 6.1.2

Factor $\cos (x)$ out of $- \cos (x)$ .

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

Step 6.1.3

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

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

Step 6.2

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

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

Step 6.3

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

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

Step 6.4

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

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

Step 6.5

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

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

Step 6.6

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

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

Step 6.7

Apply pythagorean identity.

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

Step 6.8

Rewrite using the commutative property of multiplication.

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

Step 6.9

Multiply $\cos (x)$ by $\cos^{2} (x)$ by adding the exponents.

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Step 6.9.1

Move $\cos^{2} (x)$ .

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

Step 6.9.2

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

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Step 6.9.2.1

Raise $\cos (x)$ to the power of $1$ .

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

Step 6.9.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 6.9.3

Add $2$ and $1$ .

$- \cos^{3} (x) = 0$

Step 7

Solve for $x$ .

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Step 7.1

Divide each term in $- \cos^{3} (x) = 0$ by $- 1$ and simplify.

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Step 7.1.1

Divide each term in $- \cos^{3} (x) = 0$ by $- 1$ .

$\frac{- \cos^{3} (x)}{- 1} = \frac{0}{- 1}$

Step 7.1.2

Simplify the left side.

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Step 7.1.2.1

Dividing two negative values results in a positive value.

$\frac{\cos^{3} (x)}{1} = \frac{0}{- 1}$

Step 7.1.2.2

Divide $\cos^{3} (x)$ by $1$ .

$\cos^{3} (x) = \frac{0}{- 1}$

Step 7.1.3

Simplify the right side.

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Step 7.1.3.1

Divide $0$ by $- 1$ .

$\cos^{3} (x) = 0$

Step 7.2

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

$\cos (x) = \sqrt[3]{0}$

Step 7.3

Simplify $\sqrt[3]{0}$ .

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Step 7.3.1

Rewrite $0$ as $0^{3}$ .

$\cos (x) = \sqrt[3]{0^{3}}$

Step 7.3.2

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

$\cos (x) = 0$

Step 7.4

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

$x = \arccos (0)$

Step 7.5

Simplify the right side.

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Step 7.5.1

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

$x = \frac{π}{2}$

Step 7.6

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 7.7

Simplify $2 π - \frac{π}{2}$ .

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Step 7.7.1

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 7.7.2

Combine fractions.

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Step 7.7.2.1

Combine $2 π$ and $\frac{2}{2}$ .

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

Step 7.7.2.2

Combine the numerators over the common denominator.

$x = \frac{2 π \cdot 2 - π}{2}$

Step 7.7.3

Simplify the numerator.

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Step 7.7.3.1

Multiply $2$ by $2$ .

$x = \frac{4 π - π}{2}$

Step 7.7.3.2

Subtract $π$ from $4 π$ .

$x = \frac{3 π}{2}$

Step 7.8

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

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Step 7.8.1

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

$\frac{2 π}{| b |}$

Step 7.8.2

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

$\frac{2 π}{| 1 |}$

Step 7.8.3

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

$\frac{2 π}{1}$

Step 7.8.4

Divide $2 π$ by $1$ .

$2 π$

Step 7.9

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 8

Consolidate the answers.

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