Algebra Examples

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

Step 1

Separate fractions.

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

Step 2

Convert from $\frac{1}{\cos (y)}$ to $\sec (y)$ .

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

Step 3

Simplify the expression.

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Step 3.1

Divide ${(\sin (y) - \cos (y))}^{2}$ by $1$ .

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

Step 3.2

Rewrite ${(\sin (y) - \cos (y))}^{2}$ as $(\sin (y) - \cos (y)) (\sin (y) - \cos (y))$ .

$(\sin (y) - \cos (y)) (\sin (y) - \cos (y)) \sec (y)$

Step 4

Expand $(\sin (y) - \cos (y)) (\sin (y) - \cos (y))$ using the FOIL Method.

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Step 4.1

Apply the distributive property.

$(\sin (y) (\sin (y) - \cos (y)) - \cos (y) (\sin (y) - \cos (y))) \sec (y)$

Step 4.2

Apply the distributive property.

$(\sin (y) \sin (y) + \sin (y) (- \cos (y)) - \cos (y) (\sin (y) - \cos (y))) \sec (y)$

Step 4.3

Apply the distributive property.

$(\sin (y) \sin (y) + \sin (y) (- \cos (y)) - \cos (y) \sin (y) - \cos (y) (- \cos (y))) \sec (y)$

Step 5

Simplify and combine like terms.

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Step 5.1

Simplify each term.

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Step 5.1.1

Multiply $\sin (y) \sin (y)$ .

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Step 5.1.1.1

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

$(\sin^{1} (y) \sin (y) + \sin (y) (- \cos (y)) - \cos (y) \sin (y) - \cos (y) (- \cos (y))) \sec (y)$

Step 5.1.1.2

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

$(\sin^{1} (y) \sin^{1} (y) + \sin (y) (- \cos (y)) - \cos (y) \sin (y) - \cos (y) (- \cos (y))) \sec (y)$

Step 5.1.1.3

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

$({\sin (y)}^{1 + 1} + \sin (y) (- \cos (y)) - \cos (y) \sin (y) - \cos (y) (- \cos (y))) \sec (y)$

Step 5.1.1.4

Add $1$ and $1$ .

$(\sin^{2} (y) + \sin (y) (- \cos (y)) - \cos (y) \sin (y) - \cos (y) (- \cos (y))) \sec (y)$

Step 5.1.2

Rewrite using the commutative property of multiplication.

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

Step 5.1.3

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

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Step 5.1.3.1

Multiply $- 1$ by $- 1$ .

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

Step 5.1.3.2

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

$(\sin^{2} (y) - \sin (y) \cos (y) - \cos (y) \sin (y) + \cos (y) \cos (y)) \sec (y)$

Step 5.1.3.3

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

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

Step 5.1.3.4

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

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

Step 5.1.3.5

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

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

Step 5.1.3.6

Add $1$ and $1$ .

$(\sin^{2} (y) - \sin (y) \cos (y) - \cos (y) \sin (y) + \cos^{2} (y)) \sec (y)$

Step 5.2

Reorder the factors of $- \sin (y) \cos (y)$ .

$(\sin^{2} (y) - \cos (y) \sin (y) - \cos (y) \sin (y) + \cos^{2} (y)) \sec (y)$

Step 5.3

Subtract $\cos (y) \sin (y)$ from $- \cos (y) \sin (y)$ .

$(\sin^{2} (y) - 2 \cos (y) \sin (y) + \cos^{2} (y)) \sec (y)$

Step 6

Move $\cos^{2} (y)$ .

$(\sin^{2} (y) + \cos^{2} (y) - 2 \cos (y) \sin (y)) \sec (y)$

Step 7

Apply pythagorean identity.

$(1 - 2 \cos (y) \sin (y)) \sec (y)$

Step 8

Simplify by multiplying through.

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Step 8.1

Apply the distributive property.

$1 \sec (y) - 2 \cos (y) \sin (y) \sec (y)$

Step 8.2

Multiply $\sec (y)$ by $1$ .

$\sec (y) - 2 \cos (y) \sin (y) \sec (y)$