Trigonometry Examples

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

Step 1

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

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Step 1.1

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

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Step 1.1.1

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

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

Step 1.1.2

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

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

Step 1.1.3

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

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

Step 1.2

Apply pythagorean identity.

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

Step 1.3

Simplify terms.

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Step 1.3.1

Multiply $\sin^{2} (x) \csc^{2} (x) - 1$ by $1$ .

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

Step 1.3.2

Simplify each term.

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Step 1.3.2.1

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

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

Step 1.3.2.2

Apply the product rule to $\frac{1}{\sin (x)}$ .

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

Step 1.3.2.3

One to any power is one.

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

Step 1.3.2.4

Cancel the common factor of $\sin^{2} (x)$ .

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Step 1.3.2.4.1

Cancel the common factor.

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

Step 1.3.2.4.2

Rewrite the expression.

$1 - 1 = 0$

Step 1.3.3

Subtract $1$ from $1$ .

$0 = 0$

Step 2

Graph each side of the equation. The solution is the x-value of the point of intersection.

$x \approx 0.1, \frac{1}{5}, 0.3, \frac{2}{5}, - \frac{1}{2}, - \frac{3}{5}, - 0.7$

Step 3

The result can be shown in multiple forms.

Exact Form:

$x \approx 0.1, \frac{1}{5}, 0.3, \frac{2}{5}, - \frac{1}{2}, - \frac{3}{5}, - 0.7$

Decimal Form:

$x \approx 0.1, 0.2, 0.3, 0.4, - 0.5, - 0.6, - 0.7$

Step 4