Trigonometry Examples

$\sqrt{1 - \sin (x) \cdot \sin (x)} = \cos (x)$

Step 1

To remove the radical on the left side of the equation, square both sides of the equation.

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

Step 2

Simplify each side of the equation.

Tap for more steps...

Step 2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{1 - \sin (x) \cdot \sin (x)}$ as ${(1 - \sin (x) \cdot \sin (x))}^{\frac{1}{2}}$ .

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

Step 2.2

Simplify the left side.

Tap for more steps...

Step 2.2.1

Simplify ${({(1 - \sin (x) \cdot \sin (x))}^{\frac{1}{2}})}^{2}$ .

Tap for more steps...

Step 2.2.1.1

Simplify terms.

Tap for more steps...

Step 2.2.1.1.1

Multiply the exponents in ${({(1 - \sin (x) \cdot \sin (x))}^{\frac{1}{2}})}^{2}$ .

Tap for more steps...

Step 2.2.1.1.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 2.2.1.1.1.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.2.1.1.1.2.1

Cancel the common factor.

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

Step 2.2.1.1.1.2.2

Rewrite the expression.

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

Step 2.2.1.1.2

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

Tap for more steps...

Step 2.2.1.1.2.1

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

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

Step 2.2.1.1.2.2

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

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

Step 2.2.1.1.2.3

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

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

Step 2.2.1.1.2.4

Add $1$ and $1$ .

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

Step 2.2.1.2

Apply pythagorean identity.

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

Step 2.2.1.3

Multiply the exponents in ${(\cos^{2} (x))}^{1}$ .

Tap for more steps...

Step 2.2.1.3.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 2.2.1.3.2

Multiply $2$ by $1$ .

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

Step 3

Solve for $x$ .

Tap for more steps...

Step 3.1

Since the exponents are equal, the bases of the exponents on both sides of the equation must be equal.

$| \cos (x) | = | \cos (x) |$

Step 3.2

Solve for $x$ .

Tap for more steps...

Step 3.2.1

Rewrite the absolute value equation as four equations without absolute value bars.

$\cos (x) = \cos (x)$

$\cos (x) = - \cos (x)$

$- \cos (x) = \cos (x)$

$- \cos (x) = - \cos (x)$

Step 3.2.2

After simplifying, there are only two unique equations to be solved.

$\cos (x) = \cos (x)$

$\cos (x) = - \cos (x)$

Step 3.2.3

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

Tap for more steps...

Step 3.2.3.1

For the two functions to be equal, the arguments of each must be equal.

$x = x$

Step 3.2.3.2

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

Tap for more steps...

Step 3.2.3.2.1

Subtract $x$ from both sides of the equation.

$x - x = 0$

Step 3.2.3.2.2

Subtract $x$ from $x$ .

$0 = 0$

Step 3.2.3.3

Since $0 = 0$ , the equation will always be true.

All real numbers

Step 3.2.4

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

Tap for more steps...

Step 3.2.4.1

Move all terms containing $\cos (x)$ to the left side of the equation.

Tap for more steps...

Step 3.2.4.1.1

Add $\cos (x)$ to both sides of the equation.

$\cos (x) + \cos (x) = 0$

Step 3.2.4.1.2

Add $\cos (x)$ and $\cos (x)$ .

$2 \cos (x) = 0$

Step 3.2.4.2

Divide each term in $2 \cos (x) = 0$ by $2$ and simplify.

Tap for more steps...

Step 3.2.4.2.1

Divide each term in $2 \cos (x) = 0$ by $2$ .

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

Step 3.2.4.2.2

Simplify the left side.

Tap for more steps...

Step 3.2.4.2.2.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.2.4.2.2.1.1

Cancel the common factor.

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

Step 3.2.4.2.2.1.2

Divide $\cos (x)$ by $1$ .

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

Step 3.2.4.2.3

Simplify the right side.

Tap for more steps...

Step 3.2.4.2.3.1

Divide $0$ by $2$ .

$\cos (x) = 0$

Step 3.2.4.3

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

$x = \arccos (0)$

Step 3.2.4.4

Simplify the right side.

Tap for more steps...

Step 3.2.4.4.1

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

$x = \frac{π}{2}$

Step 3.2.4.5

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 3.2.4.6

Simplify $2 π - \frac{π}{2}$ .

Tap for more steps...

Step 3.2.4.6.1

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 3.2.4.6.2

Combine fractions.

Tap for more steps...

Step 3.2.4.6.2.1

Combine $2 π$ and $\frac{2}{2}$ .

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

Step 3.2.4.6.2.2

Combine the numerators over the common denominator.

$x = \frac{2 π \cdot 2 - π}{2}$

Step 3.2.4.6.3

Simplify the numerator.

Tap for more steps...

Step 3.2.4.6.3.1

Multiply $2$ by $2$ .

$x = \frac{4 π - π}{2}$

Step 3.2.4.6.3.2

Subtract $π$ from $4 π$ .

$x = \frac{3 π}{2}$

Step 3.2.4.7

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

Tap for more steps...

Step 3.2.4.7.1

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

$\frac{2 π}{| b |}$

Step 3.2.4.7.2

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

$\frac{2 π}{| 1 |}$

Step 3.2.4.7.3

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

$\frac{2 π}{1}$

Step 3.2.4.7.4

Divide $2 π$ by $1$ .

$2 π$

Step 3.2.4.8

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 4

Consolidate the answers.

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