Trigonometry Examples

$\frac{\sec^{2} (t)}{\tan (t)} = \cot (t) + \tan (t)$

Step 1

Start on the left side.

$\frac{\sec^{2} (t)}{\tan (t)}$

Step 2

Apply Pythagorean identity in reverse.

$\frac{\tan^{2} (t) + 1}{\tan (t)}$

Step 3

Convert to sines and cosines.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

Apply the product rule to $\frac{\sin (t)}{\cos (t)}$ .

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

Step 4

Simplify.

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Step 4.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 4.2

Apply the distributive property.

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

Step 4.3

Combine.

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

Step 4.4

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

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

Step 4.5

Simplify each term.

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

Step 5

Add fractions.

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Step 5.1

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

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

Step 5.2

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

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

Step 5.3

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

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Step 5.3.1

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

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

Step 5.3.2

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

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

Step 5.3.3

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

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

Step 5.4

Combine the numerators over the common denominator.

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

Step 6

Simplify each term.

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

Step 7

Reorder terms.

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

Step 8

Now consider the right side of the equation.

$\cot (t) + \tan (t)$

Step 9

Convert to sines and cosines.

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Step 9.1

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

$\frac{\cos (t)}{\sin (t)} + \tan (t)$

Step 9.2

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

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

Step 10

Add fractions.

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Step 10.1

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

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

Step 10.2

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

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

Step 10.3

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

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Step 10.3.1

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

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

Step 10.3.2

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

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

Step 10.3.3

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

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

Step 10.4

Combine the numerators over the common denominator.

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

Step 11

Simplify each term.

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

Step 12

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

$\frac{\sec^{2} (t)}{\tan (t)} = \cot (t) + \tan (t)$ is an identity