Trigonometry Examples

$\cos^{2} (x) \csc (x) \sec (x) = \cot (x)$

Step 1

Simplify the left side.

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Step 1.1

Simplify $\cos^{2} (x) \csc (x) \sec (x)$ .

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Step 1.1.1

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

$\cos^{2} (x) \frac{1}{\sin (x)} \sec (x) = \cot (x)$

Step 1.1.2

Combine $\cos^{2} (x)$ and $\frac{1}{\sin (x)}$ .

$\frac{\cos^{2} (x)}{\sin (x)} \sec (x) = \cot (x)$

Step 1.1.3

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

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

Step 1.1.4

Combine.

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

Step 1.1.5

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

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Step 1.1.5.1

Factor $\cos (x)$ out of $\cos^{2} (x) \cdot 1$ .

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

Step 1.1.5.2

Cancel the common factors.

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Step 1.1.5.2.1

Factor $\cos (x)$ out of $\sin (x) \cos (x)$ .

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

Step 1.1.5.2.2

Cancel the common factor.

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

Step 1.1.5.2.3

Rewrite the expression.

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

Step 1.1.6

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

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

Step 2

Simplify the right side.

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Step 2.1

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

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

Step 3

Multiply both sides of the equation by $\sin (x)$ .

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

Step 4

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

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Step 4.1

Cancel the common factor.

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

Step 4.2

Rewrite the expression.

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

Step 5

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

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Step 5.1

Cancel the common factor.

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

Step 5.2

Rewrite the expression.

$\cos (x) = \cos (x)$

Step 6

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

$x = x$

Step 7

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

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Step 7.1

Subtract $x$ from both sides of the equation.

$x - x = 0$

Step 7.2

Subtract $x$ from $x$ .

$0 = 0$

Step 8

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

All real numbers

Step 9

The result can be shown in multiple forms.

All real numbers

Interval Notation:

$(- \infty, \infty)$