Algebra Examples

${(1 - \tan (x))}^{2} = \sec^{2} (x) - 2 \tan (x)$

Step 1

Simplify the left side.

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Step 1.1

Simplify ${(1 - \tan (x))}^{2}$ .

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Step 1.1.1

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

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

Step 1.1.2

Rewrite ${(1 - \frac{\sin (x)}{\cos (x)})}^{2}$ as $(1 - \frac{\sin (x)}{\cos (x)}) (1 - \frac{\sin (x)}{\cos (x)})$ .

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

Step 1.1.3

Expand $(1 - \frac{\sin (x)}{\cos (x)}) (1 - \frac{\sin (x)}{\cos (x)})$ using the FOIL Method.

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Step 1.1.3.1

Apply the distributive property.

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

Step 1.1.3.2

Apply the distributive property.

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

Step 1.1.3.3

Apply the distributive property.

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

Step 1.1.4

Simplify and combine like terms.

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Step 1.1.4.1

Simplify each term.

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Step 1.1.4.1.1

Multiply $1$ by $1$ .

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

Step 1.1.4.1.2

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

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

Step 1.1.4.1.3

Multiply $- 1$ by $1$ .

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

Step 1.1.4.1.4

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

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Step 1.1.4.1.4.1

Multiply $- 1$ by $- 1$ .

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

Step 1.1.4.1.4.2

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

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

Step 1.1.4.1.4.3

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

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

Step 1.1.4.1.4.4

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

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

Step 1.1.4.1.4.5

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

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

Step 1.1.4.1.4.6

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

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

Step 1.1.4.1.4.7

Add $1$ and $1$ .

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

Step 1.1.4.1.4.8

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

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

Step 1.1.4.1.4.9

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

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

Step 1.1.4.1.4.10

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

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

Step 1.1.4.1.4.11

Add $1$ and $1$ .

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

Step 1.1.4.2

Subtract $\frac{\sin (x)}{\cos (x)}$ from $- \frac{\sin (x)}{\cos (x)}$ .

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

Step 1.1.5

Simplify each term.

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Step 1.1.5.1

Combine $- 2$ and $\frac{\sin (x)}{\cos (x)}$ .

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

Step 1.1.5.2

Move the negative in front of the fraction.

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

Step 2

Simplify the right side.

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Step 2.1

Simplify each term.

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Step 2.1.1

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

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

Step 2.1.2

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

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

Step 2.1.3

One to any power is one.

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

Step 2.1.4

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

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

Step 2.1.5

Combine $- 2$ and $\frac{\sin (x)}{\cos (x)}$ .

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

Step 2.1.6

Move the negative in front of the fraction.

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

Step 3

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

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

Step 4

Apply the distributive property.

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

Step 5

Simplify.

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Step 5.1

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

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

Step 5.2

Rewrite using the commutative property of multiplication.

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

Step 5.3

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

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Step 5.3.1

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

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

Step 5.3.2

Cancel the common factor.

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

Step 5.3.3

Rewrite the expression.

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

Step 6

Simplify each term.

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Step 6.1

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

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Step 6.1.1

Factor $\cos (x)$ out of $- \cos (x)$ .

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

Step 6.1.2

Cancel the common factor.

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

Step 6.1.3

Rewrite the expression.

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

Step 6.2

Multiply $2$ by $- 1$ .

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

Step 7

Apply the distributive property.

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

Step 8

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

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Step 8.1

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

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

Step 8.2

Cancel the common factor.

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

Step 8.3

Rewrite the expression.

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

Step 9

Rewrite using the commutative property of multiplication.

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

Step 10

Simplify each term.

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Step 10.1

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

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Step 10.1.1

Factor $\cos (x)$ out of $- \cos (x)$ .

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

Step 10.1.2

Cancel the common factor.

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

Step 10.1.3

Rewrite the expression.

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

Step 10.2

Multiply $2$ by $- 1$ .

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

Step 11

Move all the expressions to the left side of the equation.

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Step 11.1

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

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

Step 11.2

Add $2 \sin (x)$ to both sides of the equation.

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

Step 12

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

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Step 12.1

Combine the opposite terms in $\cos (x) - 2 \sin (x) + \frac{\sin^{2} (x)}{\cos (x)} - \frac{1}{\cos (x)} + 2 \sin (x)$ .

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Step 12.1.1

Add $- 2 \sin (x)$ and $2 \sin (x)$ .

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

Step 12.1.2

Add $\cos (x)$ and $0$ .

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

Step 12.2

Combine the numerators over the common denominator.

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

Step 12.3

Reorder $\sin^{2} (x)$ and $- 1$ .

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

Step 12.4

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

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

Step 12.5

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

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

Step 12.6

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

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

Step 12.7

Rewrite $- 1 (1 - \sin^{2} (x))$ as $- (1 - \sin^{2} (x))$ .

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

Step 12.8

Apply pythagorean identity.

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

Step 12.9

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

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Step 12.9.1

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

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

Step 12.9.2

Cancel the common factors.

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Step 12.9.2.1

Multiply by $1$ .

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

Step 12.9.2.2

Cancel the common factor.

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

Step 12.9.2.3

Rewrite the expression.

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

Step 12.9.2.4

Divide $- \cos (x)$ by $1$ .

$\cos (x) - \cos (x) = 0$

Step 12.10

Subtract $\cos (x)$ from $\cos (x)$ .

$0 = 0$

Step 13

Since $0 = 0$ , the equation will always be true.

Always true

Step 14

The result can be shown in multiple forms.

Always true

Interval Notation:

$(- \infty, \infty)$