Trigonometry Examples

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

Step 1

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

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Step 1.1

Simplify each term.

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Step 1.1.1

Apply pythagorean identity.

$\frac{\csc^{2} (x)}{\cot^{2} (x)} - 1 = \tan^{2} (x)$

Step 1.1.2

Rewrite $\frac{\csc^{2} (x)}{\cot^{2} (x)}$ as ${(\frac{\csc (x)}{\cot (x)})}^{2}$ .

${(\frac{\csc (x)}{\cot (x)})}^{2} - 1 = \tan^{2} (x)$

Step 1.1.3

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

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

Step 1.1.4

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

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

Step 1.1.5

Multiply by the reciprocal of the fraction to divide by $\frac{\cos (x)}{\sin (x)}$ .

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

Step 1.1.6

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

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Step 1.1.6.1

Cancel the common factor.

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

Step 1.1.6.2

Rewrite the expression.

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

Step 1.1.7

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

$\sec^{2} (x) - 1 = \tan^{2} (x)$

Step 1.2

Apply pythagorean identity.

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

Step 2

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

$x = x$

Step 3

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

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Step 3.1

Subtract $x$ from both sides of the equation.

$x - x = 0$

Step 3.2

Subtract $x$ from $x$ .

$0 = 0$

Step 4

Since $0 = 0$ , the equation will always be true for any value of $x$ .

All real numbers

Step 5

The result can be shown in multiple forms.

All real numbers

Interval Notation:

$(- \infty, \infty)$