Trigonometry Examples

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

Step 1

Simplify the left side.

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Step 1.1

Simplify each term.

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Step 1.1.1

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

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

Step 1.1.2

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

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

Step 2

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

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

Step 3

Apply the distributive property.

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

Step 4

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

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Step 4.1

Cancel the common factor.

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

Step 4.2

Rewrite the expression.

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

Step 5

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

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Step 5.1

Cancel the common factor.

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

Step 5.2

Rewrite the expression.

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

Step 6

Multiply $\cos (x) \frac{\cos (x)}{1 - \sin (x)}$ .

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Step 6.1

Combine $\cos (x)$ and $\frac{\cos (x)}{1 - \sin (x)}$ .

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

Step 6.2

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

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

Step 6.3

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

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

Step 6.4

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

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

Step 6.5

Add $1$ and $1$ .

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

Step 7

Subtract $\frac{\cos^{2} (x)}{1 - \sin (x)}$ from both sides of the equation.

$1 + \sin (x) - \frac{\cos^{2} (x)}{1 - \sin (x)} = 0$

Step 8

Simplify $1 + \sin (x) - \frac{\cos^{2} (x)}{1 - \sin (x)}$ .

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Step 8.1

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

$1 + \frac{\sin (x) (1 - \sin (x))}{1 - \sin (x)} - \frac{\cos^{2} (x)}{1 - \sin (x)} = 0$

Step 8.2

Combine the numerators over the common denominator.

$1 + \frac{\sin (x) (1 - \sin (x)) - \cos^{2} (x)}{1 - \sin (x)} = 0$

Step 8.3

Simplify each term.

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Step 8.3.1

Simplify the numerator.

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Step 8.3.1.1

Apply the distributive property.

$1 + \frac{\sin (x) \cdot 1 + \sin (x) (- \sin (x)) - \cos^{2} (x)}{1 - \sin (x)} = 0$

Step 8.3.1.2

Multiply $\sin (x)$ by $1$ .

$1 + \frac{\sin (x) + \sin (x) (- \sin (x)) - \cos^{2} (x)}{1 - \sin (x)} = 0$

Step 8.3.1.3

Rewrite using the commutative property of multiplication.

$1 + \frac{\sin (x) - \sin (x) \sin (x) - \cos^{2} (x)}{1 - \sin (x)} = 0$

Step 8.3.1.4

Multiply $- \sin (x) \sin (x)$ .

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Step 8.3.1.4.1

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

$1 + \frac{\sin (x) - (\sin^{1} (x) \sin (x)) - \cos^{2} (x)}{1 - \sin (x)} = 0$

Step 8.3.1.4.2

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

$1 + \frac{\sin (x) - (\sin^{1} (x) \sin^{1} (x)) - \cos^{2} (x)}{1 - \sin (x)} = 0$

Step 8.3.1.4.3

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

$1 + \frac{\sin (x) - {\sin (x)}^{1 + 1} - \cos^{2} (x)}{1 - \sin (x)} = 0$

Step 8.3.1.4.4

Add $1$ and $1$ .

$1 + \frac{\sin (x) - \sin^{2} (x) - \cos^{2} (x)}{1 - \sin (x)} = 0$

Step 8.3.1.5

Factor $- 1$ out of $- \sin^{2} (x)$ .

$1 + \frac{\sin (x) - (\sin^{2} (x)) - \cos^{2} (x)}{1 - \sin (x)} = 0$

Step 8.3.1.6

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

$1 + \frac{\sin (x) - (\sin^{2} (x)) - (\cos^{2} (x))}{1 - \sin (x)} = 0$

Step 8.3.1.7

Factor $- 1$ out of $- (\sin^{2} (x)) - (\cos^{2} (x))$ .

$1 + \frac{\sin (x) - (\sin^{2} (x) + \cos^{2} (x))}{1 - \sin (x)} = 0$

Step 8.3.1.8

Apply pythagorean identity.

$1 + \frac{\sin (x) - 1 \cdot 1}{1 - \sin (x)} = 0$

Step 8.3.1.9

Multiply $- 1$ by $1$ .

$1 + \frac{\sin (x) - 1}{1 - \sin (x)} = 0$

Step 8.3.2

Cancel the common factor of $\sin (x) - 1$ and $1 - \sin (x)$ .

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Step 8.3.2.1

Factor $- 1$ out of $\sin (x)$ .

$1 + \frac{- 1 (- \sin (x)) - 1}{1 - \sin (x)} = 0$

Step 8.3.2.2

Rewrite $- 1$ as $- 1 (1)$ .

$1 + \frac{- 1 (- \sin (x)) - 1 (1)}{1 - \sin (x)} = 0$

Step 8.3.2.3

Factor $- 1$ out of $- 1 (- \sin (x)) - 1 (1)$ .

$1 + \frac{- 1 (- \sin (x) + 1)}{1 - \sin (x)} = 0$

Step 8.3.2.4

Reorder terms.

$1 + \frac{- 1 (1 - \sin (x))}{1 - \sin (x)} = 0$

Step 8.3.2.5

Cancel the common factor.

$1 + \frac{- 1 (1 - \sin (x))}{1 - \sin (x)} = 0$

Step 8.3.2.6

Divide $- 1$ by $1$ .

$1 - 1 = 0$

Step 8.4

Subtract $1$ from $1$ .

$0 = 0$

Step 9

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

All real numbers

Step 10

The result can be shown in multiple forms.

All real numbers

Interval Notation:

$(- \infty, \infty)$