Trigonometry Examples

$\tan (x) + 2 \cos (x) \csc (x) = \sec (x) \csc (x) + \cot (x)$

Step 1

Start on the left side.

$\tan (x) + 2 \cos (x) \csc (x)$

Step 2

Simplify each term.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

Multiply $2 \cos (x) \frac{1}{\sin (x)}$ .

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Step 2.3.1

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

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

Step 2.3.2

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

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

Step 3

Add fractions.

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Step 3.1

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

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

Step 3.2

To write $\frac{2 \cos (x)}{\sin (x)}$ as a fraction with a common denominator, multiply by $\frac{\cos (x)}{\cos (x)}$ .

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

Step 3.3

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

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Step 3.3.1

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

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

Step 3.3.2

Multiply $\frac{2 \cos (x)}{\sin (x)}$ by $\frac{\cos (x)}{\cos (x)}$ .

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

Step 3.3.3

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

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

Step 3.4

Combine the numerators over the common denominator.

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

Step 4

Simplify each term.

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

Step 5

Apply Pythagorean identity in reverse.

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

Step 6

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

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

Step 7

Now consider the right side of the equation.

$\sec (x) \csc (x) + \cot (x)$

Step 8

Convert to sines and cosines.

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Step 8.1

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

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

Step 8.2

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

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

Step 8.3

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

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

Step 9

Multiply $\frac{1}{\cos (x)}$ by $\frac{1}{\sin (x)}$ .

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

Step 10

Add fractions.

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Step 10.1

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

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

Step 10.2

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

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Step 10.2.1

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

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

Step 10.2.2

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

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

Step 10.3

Combine the numerators over the common denominator.

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

Step 11

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

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

Step 12

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

$\tan (x) + 2 \cos (x) \csc (x) = \sec (x) \csc (x) + \cot (x)$ is an identity