Trigonometry Examples

$\frac{\sec (t) - \cos (t)}{3 \tan (t) \sin (t)}$

Step 1

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

$\frac{\frac{1}{\cos (t)} - \cos (t)}{3 \tan (t) \sin (t)}$

Step 2

Simplify the denominator.

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Step 2.1

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

$\frac{\frac{1}{\cos (t)} - \cos (t)}{3 \frac{\sin (t)}{\cos (t)} \sin (t)}$

Step 2.2

Combine exponents.

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Step 2.2.1

Combine $3$ and $\frac{\sin (t)}{\cos (t)}$ .

$\frac{\frac{1}{\cos (t)} - \cos (t)}{\frac{3 \sin (t)}{\cos (t)} \sin (t)}$

Step 2.2.2

Combine $\frac{3 \sin (t)}{\cos (t)}$ and $\sin (t)$ .

$\frac{\frac{1}{\cos (t)} - \cos (t)}{\frac{3 \sin (t) \sin (t)}{\cos (t)}}$

Step 2.2.3

Raise $\sin (t)$ to the power of $1$ .

$\frac{\frac{1}{\cos (t)} - \cos (t)}{\frac{3 (\sin^{1} (t) \sin (t))}{\cos (t)}}$

Step 2.2.4

Raise $\sin (t)$ to the power of $1$ .

$\frac{\frac{1}{\cos (t)} - \cos (t)}{\frac{3 (\sin^{1} (t) \sin^{1} (t))}{\cos (t)}}$

Step 2.2.5

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{\frac{1}{\cos (t)} - \cos (t)}{\frac{3 {\sin (t)}^{1 + 1}}{\cos (t)}}$

Step 2.2.6

Add $1$ and $1$ .

$\frac{\frac{1}{\cos (t)} - \cos (t)}{\frac{3 \sin^{2} (t)}{\cos (t)}}$

Step 3

Multiply the numerator by the reciprocal of the denominator.

$(\frac{1}{\cos (t)} - \cos (t)) \frac{\cos (t)}{3 \sin^{2} (t)}$

Step 4

Apply the distributive property.

$\frac{1}{\cos (t)} \cdot \frac{\cos (t)}{3 \sin^{2} (t)} - \cos (t) \frac{\cos (t)}{3 \sin^{2} (t)}$

Step 5

Combine.

$\frac{1 \cos (t)}{\cos (t) (3 \sin^{2} (t))} - \cos (t) \frac{\cos (t)}{3 \sin^{2} (t)}$

Step 6

Multiply $- \cos (t) \frac{\cos (t)}{3 \sin^{2} (t)}$ .

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Step 6.1

Combine $\frac{\cos (t)}{3 \sin^{2} (t)}$ and $\cos (t)$ .

$\frac{1 \cos (t)}{\cos (t) (3 \sin^{2} (t))} - \frac{\cos (t) \cos (t)}{3 \sin^{2} (t)}$

Step 6.2

Raise $\cos (t)$ to the power of $1$ .

$\frac{1 \cos (t)}{\cos (t) (3 \sin^{2} (t))} - \frac{\cos^{1} (t) \cos (t)}{3 \sin^{2} (t)}$

Step 6.3

Raise $\cos (t)$ to the power of $1$ .

$\frac{1 \cos (t)}{\cos (t) (3 \sin^{2} (t))} - \frac{\cos^{1} (t) \cos^{1} (t)}{3 \sin^{2} (t)}$

Step 6.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{1 \cos (t)}{\cos (t) (3 \sin^{2} (t))} - \frac{{\cos (t)}^{1 + 1}}{3 \sin^{2} (t)}$

Step 6.5

Add $1$ and $1$ .

$\frac{1 \cos (t)}{\cos (t) (3 \sin^{2} (t))} - \frac{\cos^{2} (t)}{3 \sin^{2} (t)}$

Step 7

Simplify terms.

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Step 7.1

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

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Step 7.1.1

Cancel the common factor.

$\frac{1 \cos (t)}{\cos (t) (3 \sin^{2} (t))} - \frac{\cos^{2} (t)}{3 \sin^{2} (t)}$

Step 7.1.2

Rewrite the expression.

$\frac{1}{3 \sin^{2} (t)} - \frac{\cos^{2} (t)}{3 \sin^{2} (t)}$

Step 7.2

Combine the numerators over the common denominator.

$\frac{1 - \cos^{2} (t)}{3 \sin^{2} (t)}$

Step 8

Apply pythagorean identity.

$\frac{\sin^{2} (t)}{3 \sin^{2} (t)}$

Step 9

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

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Step 9.1

Cancel the common factor.

$\frac{\sin^{2} (t)}{3 \sin^{2} (t)}$

Step 9.2

Rewrite the expression.

$\frac{1}{3}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{3}$

Decimal Form:

$0.\overline{3}$