Trigonometry Examples

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

Step 1

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

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Step 1.1

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

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

Step 1.2

Add $1$ to both sides of the equation.

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

Step 2

Replace $\sin^{2} (x)$ with $1 - \cos^{2} (x)$ .

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

Step 3

Solve for $x$ .

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Step 3.1

Simplify the left side.

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Step 3.1.1

Apply pythagorean identity.

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

Step 3.2

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

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

Step 3.3

Add $1$ and $1$ .

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

Step 3.4

Subtract $2$ from both sides of the equation.

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

Step 3.5

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

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Step 3.5.1

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

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

Step 3.5.2

Simplify the left side.

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Step 3.5.2.1

Dividing two negative values results in a positive value.

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

Step 3.5.2.2

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

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

Step 3.5.3

Simplify the right side.

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Step 3.5.3.1

Divide $- 2$ by $- 1$ .

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

Step 3.6

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

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

Step 3.7

The complete solution is the result of both the positive and negative portions of the solution.

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Step 3.7.1

First, use the positive value of the $\pm$ to find the first solution.

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

Step 3.7.2

Next, use the negative value of the $\pm$ to find the second solution.

$\cos (x) = - \sqrt{2}$

Step 3.7.3

The complete solution is the result of both the positive and negative portions of the solution.

$\cos (x) = \sqrt{2}, - \sqrt{2}$

Step 3.8

Set up each of the solutions to solve for $x$ .

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

$\cos (x) = - \sqrt{2}$

Step 3.9

Solve for $x$ in $\cos (x) = \sqrt{2}$ .

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Step 3.9.1

The range of cosine is $- 1 \leq y \leq 1$ . Since $\sqrt{2}$ does not fall in this range, there is no solution.

No solution

Step 3.10

Solve for $x$ in $\cos (x) = - \sqrt{2}$ .

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Step 3.10.1

The range of cosine is $- 1 \leq y \leq 1$ . Since $- \sqrt{2}$ does not fall in this range, there is no solution.

No solution