Trigonometry Examples

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

Step 1

Start on the right side.

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

Step 2

Convert to sines and cosines.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 3

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

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

Step 4

Add fractions.

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Step 4.1

To write $\frac{1}{\cos (x)}$ as a fraction with a common denominator, multiply by $\frac{\sin^{2} (x)}{\sin^{2} (x)}$ .

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

Step 4.2

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

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

Step 4.3

Write each expression with a common denominator of $\cos (x) \sin^{2} (x)$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.3.1

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

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

Step 4.3.2

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

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

Step 4.3.3

Reorder the factors of $\cos (x) \sin^{2} (x)$ .

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

Step 4.4

Combine the numerators over the common denominator.

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

Step 5

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

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

Step 6

Apply pythagorean identity.

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

Step 7

Rewrite $\frac{1}{\sin^{2} (x) \cos (x)}$ as $\csc^{2} (x) \sec (x)$ .

$\csc^{2} (x) \sec (x)$

Step 8

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

$\csc^{2} (x) \sec (x) = \sec (x) + \csc (x) \cot (x)$ is an identity