Trigonometry Examples

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

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

Step 1

Start on the left side.

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

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

Step 2

Multiply

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

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

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

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

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

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

Step 3

Combine.

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

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

Step 4

Simplify numerator.

Tap for more steps...

Step 4.1

Expand

(1 - sin (2 x)) (- sin (x) + cos (x))

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

Tap for more steps...

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))}

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

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))}

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

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))}

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

\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))}

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

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))}

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

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

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

Step 5

Simplify denominator.

Tap for more steps...

Step 5.1

Expand

(sin (x) - cos (x)) (- sin (x) + cos (x))

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

Tap for more steps...

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))}

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

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))}

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

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)}

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

\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)}

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

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)}

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

\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)}

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

Step 6

Simplify the expression.

Tap for more steps...

Step 6.1

Move

- {cos}^{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)}

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

Step 6.2

Factor

- 1

$- 1$ out of

- {sin}^{2} (x)

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

\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)}

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

Step 6.3

Factor

- 1

$- 1$ out of

- {cos}^{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)}

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

Step 6.4

Factor

- 1

$- 1$ out of

- ({sin}^{2} (x)) - ({cos}^{2} (x))

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

\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)}

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

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)}

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

Step 6.6

Simplify the numerator.

Tap for more steps...

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)}

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

Step 6.6.2

Factor out the greatest common factor from each group.

Tap for more steps...

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)}

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

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)}

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

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

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

Step 6.6.3

Factor the polynomial by factoring out the greatest common factor,

sin (x) - cos (x)

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

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

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

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

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

Step 6.7

Simplify the denominator.

Tap for more steps...

Step 6.7.1

Multiply

- 1

$- 1$ by

1

$1$ .

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

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

Step 6.7.2

Reorder

2 cos (x)

$2 \cos (x)$ and

sin (x)

$\sin (x)$ .

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

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

Step 6.7.3

Reorder

sin (x)

$\sin (x)$ and

2

$2$ .

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

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

Step 6.7.4

Apply the sine double-angle identity.

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

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

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

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

Step 6.8

Cancel the common factor of

sin (2 x) - 1

$\sin (2 x) - 1$ and

- 1 + sin (2 x)

$- 1 + \sin (2 x)$ .

Tap for more steps...

Step 6.8.1

Reorder terms.

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

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

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