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

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

Simplify the numerator.

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Apply the sine double–angle identity.

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

Use the double–angle identity to transform $\cos (2 x)$ to $\cos^{2} (x) - \sin^{2} (x)$ .

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

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

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

Apply pythagorean identity.

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

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

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

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

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Factor $2 \cos (x)$ out of $2 \sin (x) \cos (x)$ .

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

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

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

Factor $2 \cos (x)$ out of $2 \cos (x) (\sin (x)) + 2 \cos (x) (\cos (x))$ .

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

Simplify the denominator.

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Apply the sine double–angle identity.

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

Use the double–angle identity to transform $\cos (2 x)$ to $\cos^{2} (x) - \sin^{2} (x)$ .

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

Apply the distributive property.

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

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

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Multiply $- 1$ by $- 1$ .

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

Multiply $\sin^{2} (x)$ by $1$ .

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

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

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

Apply pythagorean identity.

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

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

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

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

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Factor $2 \sin (x)$ out of $2 \sin (x) \cos (x)$ .

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

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

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

Factor $2 \sin (x)$ out of $2 \sin (x) (\cos (x)) + 2 \sin (x) (\sin (x))$ .

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

Cancel the common factor of $2$ .

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Cancel the common factor.

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

Rewrite the expression.

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

Separate fractions.

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

Convert from $\frac{\cos (x)}{\sin (x)}$ to $\cot (x)$ .

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

Combine $\frac{\sin (x) + \cos (x)}{\cos (x) + \sin (x)}$ and $\cot (x)$ .

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

Reorder factors in $\frac{(\sin (x) + \cos (x)) \cot (x)}{\cos (x) + \sin (x)}$ .

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

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

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