Trigonometry Examples

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

Step 1

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

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Step 1.1

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

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

Step 1.2

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

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

Step 2

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

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

Rearrange terms.

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

Step 2.5

Apply pythagorean identity.

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

Step 2.6

Simplify each term.

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Step 2.6.1

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

$(\cos (x) - \sin (x)) (\cos (x) - \sin (x)) - 1 \cdot 1 = 0$

Step 2.6.2

Expand $(\cos (x) - \sin (x)) (\cos (x) - \sin (x))$ using the FOIL Method.

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Step 2.6.2.1

Apply the distributive property.

$\cos (x) (\cos (x) - \sin (x)) - \sin (x) (\cos (x) - \sin (x)) - 1 \cdot 1 = 0$

Step 2.6.2.2

Apply the distributive property.

$\cos (x) \cos (x) + \cos (x) (- \sin (x)) - \sin (x) (\cos (x) - \sin (x)) - 1 \cdot 1 = 0$

Step 2.6.2.3

Apply the distributive property.

$\cos (x) \cos (x) + \cos (x) (- \sin (x)) - \sin (x) \cos (x) - \sin (x) (- \sin (x)) - 1 \cdot 1 = 0$

Step 2.6.3

Simplify and combine like terms.

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Step 2.6.3.1

Simplify each term.

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Step 2.6.3.1.1

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

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Step 2.6.3.1.1.1

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

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

Step 2.6.3.1.1.2

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

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

Step 2.6.3.1.1.3

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

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

Step 2.6.3.1.1.4

Add $1$ and $1$ .

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

Step 2.6.3.1.2

Rewrite using the commutative property of multiplication.

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

Step 2.6.3.1.3

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

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Step 2.6.3.1.3.1

Multiply $- 1$ by $- 1$ .

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

Step 2.6.3.1.3.2

Multiply $\sin (x)$ by $1$ .

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

Step 2.6.3.1.3.3

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

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

Step 2.6.3.1.3.4

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

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

Step 2.6.3.1.3.5

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

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

Step 2.6.3.1.3.6

Add $1$ and $1$ .

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

Step 2.6.3.2

Reorder the factors of $- \sin (x) \cos (x)$ .

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

Step 2.6.3.3

Subtract $\cos (x) \sin (x)$ from $- \cos (x) \sin (x)$ .

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

Step 2.6.4

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

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

Step 2.6.5

Rearrange terms.

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

Step 2.6.6

Apply pythagorean identity.

$1 - 2 \cos (x) \sin (x) - 1 \cdot 1 = 0$

Step 2.6.7

Multiply $- 1$ by $1$ .

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

Step 2.7

Combine the opposite terms in $1 - 2 \cos (x) \sin (x) - 1$ .

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Step 2.7.1

Subtract $1$ from $1$ .

$- 2 \cos (x) \sin (x) + 0 = 0$

Step 2.7.2

Add $- 2 \cos (x) \sin (x)$ and $0$ .

$- 2 \cos (x) \sin (x) = 0$

Step 3

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$\cos (x) = 0$

$\sin (x) = 0$

Step 4

Set $\cos (x)$ equal to $0$ and solve for $x$ .

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Step 4.1

Set $\cos (x)$ equal to $0$ .

$\cos (x) = 0$

Step 4.2

Solve $\cos (x) = 0$ for $x$ .

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Step 4.2.1

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

$x = \arccos (0)$

Step 4.2.2

Simplify the right side.

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Step 4.2.2.1

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

$x = \frac{π}{2}$

Step 4.2.3

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 4.2.4

Simplify $2 π - \frac{π}{2}$ .

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Step 4.2.4.1

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 4.2.4.2

Combine fractions.

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Step 4.2.4.2.1

Combine $2 π$ and $\frac{2}{2}$ .

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

Step 4.2.4.2.2

Combine the numerators over the common denominator.

$x = \frac{2 π \cdot 2 - π}{2}$

Step 4.2.4.3

Simplify the numerator.

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Step 4.2.4.3.1

Multiply $2$ by $2$ .

$x = \frac{4 π - π}{2}$

Step 4.2.4.3.2

Subtract $π$ from $4 π$ .

$x = \frac{3 π}{2}$

Step 4.2.5

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

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Step 4.2.5.1

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

$\frac{2 π}{| b |}$

Step 4.2.5.2

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

$\frac{2 π}{| 1 |}$

Step 4.2.5.3

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

$\frac{2 π}{1}$

Step 4.2.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 4.2.6

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 5

Set $\sin (x)$ equal to $0$ and solve for $x$ .

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Step 5.1

Set $\sin (x)$ equal to $0$ .

$\sin (x) = 0$

Step 5.2

Solve $\sin (x) = 0$ for $x$ .

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Step 5.2.1

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

$x = \arcsin (0)$

Step 5.2.2

Simplify the right side.

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Step 5.2.2.1

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

$x = 0$

Step 5.2.3

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 5.2.4

Subtract $0$ from $π$ .

$x = π$

Step 5.2.5

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

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Step 5.2.5.1

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

$\frac{2 π}{| b |}$

Step 5.2.5.2

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

$\frac{2 π}{| 1 |}$

Step 5.2.5.3

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

$\frac{2 π}{1}$

Step 5.2.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 5.2.6

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 6

The final solution is all the values that make $- 2 \cos (x) \sin (x) = 0$ true.

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

Step 7

Consolidate the answers.

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