Trigonometry Examples

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

Step 1

Simplify the left side.

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Step 1.1

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

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Step 1.1.1

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

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Step 1.1.1.1

Factor $\tan (t)$ out of $\sec (t) \tan (t)$ .

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

Step 1.1.1.2

Cancel the common factors.

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Step 1.1.1.2.1

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

$\frac{\tan (t) \sec (t)}{\tan (t) (\tan (t) t)} = \csc (t)$

Step 1.1.1.2.2

Cancel the common factor.

$\frac{\tan (t) \sec (t)}{\tan (t) (\tan (t) t)} = \csc (t)$

Step 1.1.1.2.3

Rewrite the expression.

$\frac{\sec (t)}{\tan (t) t} = \csc (t)$

Step 1.1.2

Separate fractions.

$\frac{1}{t} \cdot \frac{\sec (t)}{\tan (t)} = \csc (t)$

Step 1.1.3

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

$\frac{1}{t} \cdot \frac{\sec (t)}{\frac{\sin (t)}{\cos (t)}} = \csc (t)$

Step 1.1.4

Rewrite $\sec (t)$ in terms of sines and cosines.

$\frac{1}{t} \cdot \frac{\frac{1}{\cos (t)}}{\frac{\sin (t)}{\cos (t)}} = \csc (t)$

Step 1.1.5

Multiply by the reciprocal of the fraction to divide by $\frac{\sin (t)}{\cos (t)}$ .

$\frac{1}{t} (\frac{1}{\cos (t)} \cdot \frac{\cos (t)}{\sin (t)}) = \csc (t)$

Step 1.1.6

Cancel the common factor of $\cos (t)$ .

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Step 1.1.6.1

Cancel the common factor.

$\frac{1}{t} (\frac{1}{\cos (t)} \cdot \frac{\cos (t)}{\sin (t)}) = \csc (t)$

Step 1.1.6.2

Rewrite the expression.

$\frac{1}{t} \cdot \frac{1}{\sin (t)} = \csc (t)$

Step 1.1.7

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

$\frac{1}{t \sin (t)} = \csc (t)$

Step 2

Simplify the right side.

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Step 2.1

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

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

Step 3

Multiply both sides of the equation by $\sin (t)$ .

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

Step 4

Cancel the common factor of $\sin (t)$ .

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Step 4.1

Factor $\sin (t)$ out of $t \sin (t)$ .

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

Step 4.2

Cancel the common factor.

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

Step 4.3

Rewrite the expression.

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

Step 5

Cancel the common factor of $\sin (t)$ .

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Step 5.1

Cancel the common factor.

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

Step 5.2

Rewrite the expression.

$\frac{1}{t} = 1$

Step 6

Find the LCD of the terms in the equation.

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Step 6.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$t, 1$

Step 6.2

The LCM of one and any expression is the expression.

$t$

Step 7

Multiply each term in $\frac{1}{t} = 1$ by $t$ to eliminate the fractions.

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Step 7.1

Multiply each term in $\frac{1}{t} = 1$ by $t$ .

$\frac{1}{t} t = 1 t$

Step 7.2

Simplify the left side.

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Step 7.2.1

Cancel the common factor of $t$ .

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Step 7.2.1.1

Cancel the common factor.

$\frac{1}{t} t = 1 t$

Step 7.2.1.2

Rewrite the expression.

$1 = 1 t$

Step 7.3

Simplify the right side.

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Step 7.3.1

Multiply $t$ by $1$ .

$1 = t$

Step 8

Rewrite the equation as $t = 1$ .

$t = 1$