Trigonometry Examples

$\frac{1}{\tan (x)} + \frac{1}{\cot (x)} = \tan (x) + \cot (x)$

Step 1

Start on the left side.

$\frac{1}{\tan (x)} + \frac{1}{\cot (x)}$

Step 2

Convert to sines and cosines.

Tap for more steps...

Step 2.1

Write $\tan (x)$ in sines and cosines using the quotient identity.

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

Step 2.2

Write $\cot (x)$ in sines and cosines using the quotient identity.

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

Step 3

Simplify each term.

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

Step 4

Add fractions.

Tap for more steps...

Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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

Tap for more steps...

Step 4.3.1

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

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

Step 4.3.2

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

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

Step 4.3.3

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

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

Step 4.4

Combine the numerators over the common denominator.

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

Step 5

Simplify each term.

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

Step 6

Now consider the right side of the equation.

$\tan (x) + \cot (x)$

Step 7

Convert to sines and cosines.

Tap for more steps...

Step 7.1

Write $\tan (x)$ in sines and cosines using the quotient identity.

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

Step 7.2

Write $\cot (x)$ in sines and cosines using the quotient identity.

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

Step 8

Add fractions.

Tap for more steps...

Step 8.1

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

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

Step 8.2

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

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

Step 8.3

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

Tap for more steps...

Step 8.3.1

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

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

Step 8.3.2

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

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

Step 8.3.3

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

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

Step 8.4

Combine the numerators over the common denominator.

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

Step 9

Simplify each term.

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

Step 10

Reorder terms.

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

Step 11

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

$\frac{1}{\tan (x)} + \frac{1}{\cot (x)} = \tan (x) + \cot (x)$ is an identity