Trigonometry Examples

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

Step 1

Start on the left side.

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

Step 2

Subtract fractions.

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Step 2.1

To write $\frac{\sec (x)}{\sin (x)}$ as a fraction with a common denominator, multiply by $\frac{\cos (x)}{\cos (x)}$ .

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

Step 2.2

To write $- \frac{\sin (x)}{\cos (x)}$ as a fraction with a common denominator, multiply by $\frac{\sin (x)}{\sin (x)}$ .

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

Step 2.3

Write each expression with a common denominator of $\sin (x) \cos (x)$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.3.1

Multiply $\frac{\sec (x)}{\sin (x)}$ by $\frac{\cos (x)}{\cos (x)}$ .

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

Step 2.3.2

Multiply $\frac{\sin (x)}{\cos (x)}$ by $\frac{\sin (x)}{\sin (x)}$ .

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

Step 2.3.3

Reorder the factors of $\sin (x) \cos (x)$ .

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

Step 2.4

Combine the numerators over the common denominator.

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

Step 3

Multiply $- \sin (x) \sin (x)$ .

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

Step 4

Apply Pythagorean identity in reverse.

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

Step 5

Apply the reciprocal identity to $\sec (x)$ .

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

Step 6

Simplify.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

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

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Step 6.1.1.1

Cancel the common factor.

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

Step 6.1.1.2

Rewrite the expression.

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

Step 6.1.2

Apply the distributive property.

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

Step 6.1.3

Multiply $- 1$ by $1$ .

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

Step 6.1.4

Multiply $- - {\cos (x)}^{2}$ .

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Step 6.1.4.1

Multiply $- 1$ by $- 1$ .

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

Step 6.1.4.2

Multiply ${\cos (x)}^{2}$ by $1$ .

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

Step 6.1.5

Subtract $1$ from $1$ .

$\frac{0 + {\cos (x)}^{2}}{\cos (x) \sin (x)}$

Step 6.1.6

Add $0$ and ${\cos (x)}^{2}$ .

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

Step 6.2

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

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

Step 7

Rewrite $\frac{\cos (x)}{\sin (x)}$ as $\cot (x)$ .

$\cot (x)$

Step 8

Because the two sides have been shown to be equivalent, the equation is an identity.

$\frac{\sec (x)}{\sin (x)} - \frac{\sin (x)}{\cos (x)} = \cot (x)$ is an identity