Trigonometry Examples

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

Step 1

Start on the right side.

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

Step 2

Convert to sines and cosines.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

Simplify.

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Step 2.3.1

Apply the product rule to $\frac{1}{\cos (x)}$ .

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

Step 2.3.2

One to any power is one.

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

Step 2.4

Simplify.

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Step 2.4.1

Apply the product rule to $\frac{1}{\sin (x)}$ .

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

Step 2.4.2

One to any power is one.

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

Step 3

Add fractions.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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Step 3.3.1

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

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

Step 3.3.2

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

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

Step 3.3.3

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

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

Step 3.4

Combine the numerators over the common denominator.

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

Step 4

Apply pythagorean identity.

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

Step 5

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

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

Step 6

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

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