Trigonometry Examples

$\tan (x) \cdot \csc (x) = \sec (x)$

Step 1

Simplify $\tan (x) \cdot \csc (x)$ .

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Step 1.1

Rewrite in terms of sines and cosines, then cancel the common factors.

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Step 1.1.1

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

$\frac{\sin (x)}{\cos (x)} \cdot \frac{1}{\sin (x)} = \sec (x)$

Step 1.1.2

Cancel the common factors.

$\frac{1}{\cos (x)} = \sec (x)$

Step 1.2

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

$\sec (x) = \sec (x)$

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:

$(- \infty, \infty)$