Trigonometry Examples

$\frac{\tan (x) \cdot \sec (x)}{\csc (x)} \cdot \cot (x) = \tan (x)$

Step 1

Simplify $\frac{\tan (x) \cdot \sec (x)}{\csc (x)} \cdot \cot (x)$ .

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Step 1.1

Separate fractions.

$\frac{\sec (x)}{1} \cdot \frac{\tan (x)}{\csc (x)} \cdot \cot (x) = \tan (x)$

Step 1.2

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

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

Step 1.3

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

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

Step 1.4

Multiply by the reciprocal of the fraction to divide by $\frac{1}{\sin (x)}$ .

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

Step 1.5

Convert from $\frac{\sin (x) \sin (x)}{\cos (x)}$ to $\sin (x) \tan (x)$ .

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

Step 1.6

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

$\sec (x) (\sin (x) \tan (x)) \cdot \cot (x) = \tan (x)$

Step 1.7

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

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

Step 1.8

Combine $\frac{1}{\cos (x)}$ and $\sin (x)$ .

$\frac{\sin (x)}{\cos (x)} \tan (x) \cdot \cot (x) = \tan (x)$

Step 1.9

Convert from $\frac{\sin (x)}{\cos (x)}$ to $\tan (x)$ .

$\tan (x) \tan (x) \cdot \cot (x) = \tan (x)$

Step 1.10

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

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Step 1.10.1

Add parentheses.

$\tan (x) (\tan (x) \cdot \cot (x)) = \tan (x)$

Step 1.10.2

Reorder $\tan (x)$ and $\cot (x)$ .

$\tan (x) (\cot (x) \cdot \tan (x)) = \tan (x)$

Step 1.10.3

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

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

Step 1.10.4

Cancel the common factors.

$\tan (x) \cdot 1 = \tan (x)$

Step 1.11

Multiply $\tan (x)$ by $1$ .

$\tan (x) = \tan (x)$

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:

$(- \infty, \infty)$