Trigonometry Examples

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

Step 1

Start on the left side.

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

Step 2

Multiply $\frac{1 - \sin (2 x)}{\sin (x) - \cos (x)}$ by $\frac{- \sin (x) + \cos (x)}{- \sin (x) + \cos (x)}$ .

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

Step 3

Combine.

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

Step 4

Simplify numerator.

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Step 4.1

Expand $(1 - \sin (2 x)) (- \sin (x) + \cos (x))$ using the FOIL Method.

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Step 4.1.1

Apply the distributive property.

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

Step 4.1.2

Apply the distributive property.

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

Step 4.1.3

Apply the distributive property.

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

Step 4.2

Simplify each term.

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

Step 5

Simplify denominator.

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Step 5.1

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

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Step 5.1.1

Apply the distributive property.

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

Step 5.1.2

Apply the distributive property.

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

Step 5.1.3

Apply the distributive property.

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

Step 5.2

Simplify and combine like terms.

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

Step 6

Simplify the expression.

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Step 6.1

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

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

Step 6.2

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

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

Step 6.3

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

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

Step 6.4

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

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

Step 6.5

Apply pythagorean identity.

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

Step 6.6

Simplify the numerator.

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Step 6.6.1

Reorder terms.

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

Step 6.6.2

Factor out the greatest common factor from each group.

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Step 6.6.2.1

Group the first two terms and the last two terms.

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

Step 6.6.2.2

Factor out the greatest common factor (GCF) from each group.

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

Step 6.6.3

Factor the polynomial by factoring out the greatest common factor, $\sin (x) - \cos (x)$ .

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

Step 6.7

Simplify the denominator.

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Step 6.7.1

Multiply $- 1$ by $1$ .

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

Step 6.7.2

Reorder $2 \cos (x)$ and $\sin (x)$ .

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

Step 6.7.3

Reorder $\sin (x)$ and $2$ .

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

Step 6.7.4

Apply the sine double-angle identity.

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

Step 6.8

Cancel the common factor of $\sin (2 x) - 1$ and $- 1 + \sin (2 x)$ .

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Step 6.8.1

Reorder terms.

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

Step 6.8.2

Cancel the common factor.

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

Step 6.8.3

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

$\sin (x) - \cos (x)$

Step 7

Because the two sides have been shown to be equivalent, the equation is an identity.

$\frac{1 - \sin (2 x)}{\sin (x) - \cos (x)} = \sin (x) - \cos (x)$ is an identity