Trigonometry Examples

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

Step 1

Start on the left side.

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

Step 2

Write $\tan (t)$ as a fraction with denominator $1$ .

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

Step 3

Add fractions.

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Step 3.1

To write $\frac{\tan (t)}{1}$ as a fraction with a common denominator, multiply by $\frac{\tan (t)}{\tan (t)}$ .

$\frac{\tan (t)}{1} \cdot \frac{\tan (t)}{\tan (t)} + \frac{1}{\tan (t)}$

Step 3.2

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

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

Step 3.3

Combine the numerators over the common denominator.

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

Step 4

Multiply $\tan (t) \tan (t)$ .

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

Step 5

Apply pythagorean identity.

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

Step 6

Convert to sines and cosines.

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Step 6.1

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

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

Step 6.2

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

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

Step 6.3

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

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

Step 7

Simplify.

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Step 7.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 7.2

Combine.

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

Step 7.3

Cancel the common factor of $\cos (t)$ and ${\cos (t)}^{2}$ .

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Step 7.3.1

Factor $\cos (t)$ out of $1^{2} \cos (t)$ .

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

Step 7.3.2

Cancel the common factors.

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Step 7.3.2.1

Factor $\cos (t)$ out of ${\cos (t)}^{2} \sin (t)$ .

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

Step 7.3.2.2

Cancel the common factor.

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

Step 7.3.2.3

Rewrite the expression.

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

Step 7.4

One to any power is one.

$\frac{1}{\cos (t) \sin (t)}$

Step 8

Rewrite $\frac{1}{\cos (t) \sin (t)}$ as $\frac{1}{\sin (t) \cos (t)}$ .

$\frac{1}{\sin (t) \cos (t)}$

Step 9

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

$\tan (t) + \frac{1}{\tan (t)} = \frac{1}{\sin (t) \cos (t)}$ is an identity