Trigonometry Examples

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

Step 1

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

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

Step 2

Simplify the numerator.

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Step 2.1

Apply the sine double-angle identity.

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

Step 2.2

Combine exponents.

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Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.2.3

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

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

Step 2.2.4

Add $1$ and $1$ .

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

Step 3

Simplify the denominator.

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Step 3.1

Use the double-angle identity to transform $\cos (2 x)$ to $2 \cos^{2} (x) - 1$ .

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

Step 3.2

Combine the opposite terms in $1 + 2 \cos^{2} (x) - 1$ .

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Step 3.2.1

Subtract $1$ from $1$ .

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

Step 3.2.2

Add $0$ and $2 \cos^{2} (x)$ .

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

Step 4

Reduce the expression by cancelling the common factors.

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Step 4.1

Cancel the common factor of $2$ .

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Step 4.1.1

Cancel the common factor.

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

Step 4.1.2

Rewrite the expression.

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

Step 4.2

Cancel the common factor of $\cos^{2} (x)$ .

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Step 4.2.1

Cancel the common factor.

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

Step 4.2.2

Rewrite the expression.

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