Trigonometry Examples

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

Step 1

Simplify the numerator.

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Step 1.1

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

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

Step 1.2

Subtract $1$ from $1$ .

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

Step 1.3

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

$\frac{2 \cos^{2} (y)}{\sin (2 y)}$

Step 2

Apply the sine double-angle identity.

$\frac{2 \cos^{2} (y)}{2 \sin (y) \cos (y)}$

Step 3

Cancel the common factor of $2$ .

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Step 3.1

Cancel the common factor.

$\frac{2 \cos^{2} (y)}{2 \sin (y) \cos (y)}$

Step 3.2

Rewrite the expression.

$\frac{\cos^{2} (y)}{\sin (y) \cos (y)}$

Step 4

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

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Step 4.1

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

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

Step 4.2

Cancel the common factors.

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Step 4.2.1

Factor $\cos (y)$ out of $\sin (y) \cos (y)$ .

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

Step 4.2.2

Cancel the common factor.

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

Step 4.2.3

Rewrite the expression.

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

Step 5

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

$\cot (y)$