Precalculus Examples

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

Step 1

Simplify the right side.

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Step 1.1

Simplify $\frac{1 - \tan^{2} (y)}{1 + \tan^{2} (y)}$ .

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Step 1.1.1

Rearrange terms.

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

Step 1.1.2

Apply pythagorean identity.

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

Step 1.1.3

Simplify the numerator.

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Step 1.1.3.1

Rewrite $1$ as $1^{2}$ .

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

Step 1.1.3.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 1$ and $b = \tan (y)$ .

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

Step 2

Since $y$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

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

Step 3

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

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Step 3.1

Subtract $\cos (2 y)$ from both sides of the equation.

$\frac{(1 + \tan (y)) (1 - \tan (y))}{\sec^{2} (y)} - \cos (2 y) = 0$

Step 3.2

Simplify each term.

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Step 3.2.1

Simplify the numerator.

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Step 3.2.1.1

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

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

Step 3.2.1.2

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

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

Step 3.2.2

Simplify the denominator.

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Step 3.2.2.1

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

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

Step 3.2.2.2

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

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

Step 3.2.2.3

One to any power is one.

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

Step 3.2.3

Multiply the numerator by the reciprocal of the denominator.

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

Step 3.2.4

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

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Step 3.2.4.1

Apply the distributive property.

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

Step 3.2.4.2

Apply the distributive property.

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

Step 3.2.4.3

Apply the distributive property.

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

Step 3.2.5

Simplify and combine like terms.

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Step 3.2.5.1

Simplify each term.

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Step 3.2.5.1.1

Multiply $1$ by $1$ .

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

Step 3.2.5.1.2

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

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

Step 3.2.5.1.3

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

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

Step 3.2.5.1.4

Rewrite using the commutative property of multiplication.

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

Step 3.2.5.1.5

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

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Step 3.2.5.1.5.1

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

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

Step 3.2.5.1.5.2

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

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

Step 3.2.5.1.5.3

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

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

Step 3.2.5.1.5.4

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

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

Step 3.2.5.1.5.5

Add $1$ and $1$ .

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

Step 3.2.5.1.5.6

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

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

Step 3.2.5.1.5.7

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

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

Step 3.2.5.1.5.8

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

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

Step 3.2.5.1.5.9

Add $1$ and $1$ .

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

Step 3.2.5.2

Add $- \frac{\sin (y)}{\cos (y)}$ and $\frac{\sin (y)}{\cos (y)}$ .

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

Step 3.2.5.3

Add $1$ and $0$ .

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

Step 3.2.6

Apply the distributive property.

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

Step 3.2.7

Multiply $\cos^{2} (y)$ by $1$ .

$\cos^{2} (y) - \frac{\sin^{2} (y)}{\cos^{2} (y)} \cos^{2} (y) - \cos (2 y) = 0$

Step 3.2.8

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

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Step 3.2.8.1

Move the leading negative in $- \frac{\sin^{2} (y)}{\cos^{2} (y)}$ into the numerator.

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

Step 3.2.8.2

Cancel the common factor.

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

Step 3.2.8.3

Rewrite the expression.

$\cos^{2} (y) - \sin^{2} (y) - \cos (2 y) = 0$

Step 3.2.9

Apply the cosine double-angle identity.

$\cos (2 y) - \cos (2 y) = 0$

Step 3.3

Subtract $\cos (2 y)$ from $\cos (2 y)$ .

$0 = 0$

Step 4

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

All real numbers

Step 5

The result can be shown in multiple forms.

All real numbers

Interval Notation:

$(- \infty, \infty)$