Trigonometry Examples

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

Step 1

Simplify the numerator.

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Step 1.1

Apply the sine double-angle identity.

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

Step 1.2

Use the double-angle identity to transform

$\cos (2 x)$ to

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

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

Step 1.3

Subtract

$1$ from

$1$ .

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

Step 1.4

Add

$0$ and

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

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

Step 1.5

Factor

$2 \cos (x)$ out of

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

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Step 1.5.1

Factor

$2 \cos (x)$ out of

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

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

Step 1.5.2

Factor

$2 \cos (x)$ out of

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

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

Step 1.5.3

Factor

$2 \cos (x)$ out of

$2 \cos (x) (\sin (x)) + 2 \cos (x) (\cos (x))$ .

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

Step 2

Simplify the denominator.

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Step 2.1

Apply the sine double-angle identity.

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

Step 2.2

Use the double-angle identity to transform

$\cos (2 x)$ to

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

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

Step 2.3

Apply the distributive property.

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

Step 2.4

Multiply

$2$ by

$- 1$ .

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

Step 2.5

Multiply

$- 1$ by

$- 1$ .

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

Step 2.6

Add

$1$ and

$1$ .

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

Step 2.7

Factor

$2$ out of

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

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Step 2.7.1

Factor

$2$ out of

$2$ .

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

Step 2.7.2

Factor

$2$ out of

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

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

Step 2.7.3

Factor

$2$ out of

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

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

Step 2.7.4

Factor

$2$ out of

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

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

Step 2.7.5

Factor

$2$ out of

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

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

Step 2.8

Move

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

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

Step 2.9

Apply pythagorean identity.

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

Step 2.10

Factor

$\sin (x)$ out of

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

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Step 2.10.1

Factor

$\sin (x)$ out of

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

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

Step 2.10.2

Factor

$\sin (x)$ out of

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

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

Step 2.10.3

Factor

$\sin (x)$ out of

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

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

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

Step 3

Reduce the expression by cancelling the common factors.

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Step 3.1

Cancel the common factor of

$2$ .

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Step 3.1.1

Cancel the common factor.

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

Step 3.1.2

Rewrite the expression.

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

Step 3.2

Cancel the common factor of

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

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Step 3.2.1

Cancel the common factor.

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

Step 3.2.2

Rewrite the expression.

$\frac{\cos (x)}{\sin (x)}$

Step 4

Convert from

$\frac{\cos (x)}{\sin (x)}$ to

$\cot (x)$ .

$\cot (x)$