Trigonometry Examples

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

Step 1

Simplify each term.

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

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

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

Step 1.4

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

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Step 1.4.1

Cancel the common factor.

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

Step 1.4.2

Rewrite the expression.

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

Step 1.5

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

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

Step 1.6

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

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

Step 1.7

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

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

Step 1.8

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

$\sec (x) - (\frac{\cos (x)}{\sin (x)} \sin (x)) = \frac{\tan (x)}{\csc (x)}$

Step 1.9

Write $\sin (x)$ as a fraction with denominator $1$ .

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

Step 1.10

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

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Step 1.10.1

Cancel the common factor.

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

Step 1.10.2

Rewrite the expression.

$\sec (x) - \cos (x) = \frac{\tan (x)}{\csc (x)}$

Step 2

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

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

$\sec (x) - \cos (x) = \sin (x) \tan (x)$

Step 3

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

$x \approx 0, 0.1, \frac{1}{5}, 0.3, - \frac{2}{5}, \frac{2}{5}, \frac{1}{2}$

Step 4