Trigonometry Examples

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

Step 1

Simplify the left side.

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Step 1.1

Simplify $\sec (t) - \frac{\cos (t)}{1 + \sin (t)}$ .

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Step 1.1.1

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

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

Step 1.1.2

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

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

Step 1.1.3

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

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

Step 1.1.4

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

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Step 1.1.4.1

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

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

Step 1.1.4.2

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

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

Step 1.1.4.3

Reorder the factors of $(1 + \sin (t)) \cos (t)$ .

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

Step 1.1.5

Combine the numerators over the common denominator.

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

Step 1.1.6

Simplify the numerator.

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Step 1.1.6.1

Multiply $- \cos (t) \cos (t)$ .

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Step 1.1.6.1.1

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

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

Step 1.1.6.1.2

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

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

Step 1.1.6.1.3

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

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

Step 1.1.6.1.4

Add $1$ and $1$ .

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

Step 1.1.6.2

Move $- \cos^{2} (t)$ .

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

Step 1.1.6.3

Apply pythagorean identity.

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

Step 1.1.6.4

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

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Step 1.1.6.4.1

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

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

Step 1.1.6.4.2

Multiply by $1$ .

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

Step 1.1.6.4.3

Factor $\sin (t)$ out of $\sin (t) \sin (t) + \sin (t) \cdot 1$ .

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

Step 1.1.7

Cancel the common factor of $\sin (t) + 1$ and $1 + \sin (t)$ .

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Step 1.1.7.1

Reorder terms.

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

Step 1.1.7.2

Cancel the common factor.

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

Step 1.1.7.3

Rewrite the expression.

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

Step 2

Simplify the right side.

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Step 2.1

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

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

Step 3

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

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

Step 4

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

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Step 4.1

Cancel the common factor.

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

Step 4.2

Rewrite the expression.

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

Step 5

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

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Step 5.1

Cancel the common factor.

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

Step 5.2

Rewrite the expression.

$\sin (t) = \sin (t)$

Step 6

For the two functions to be equal, the arguments of each must be equal.

$t = t$

Step 7

Move all terms containing $t$ to the left side of the equation.

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Step 7.1

Subtract $t$ from both sides of the equation.

$t - t = 0$

Step 7.2

Subtract $t$ from $t$ .

$0 = 0$

Step 8

Since $0 = 0$ , the equation will always be true for any value of $t$ .

All real numbers

Step 9

The result can be shown in multiple forms.

All real numbers

Interval Notation:

$(- \infty, \infty)$