Trigonometry Examples

$\frac{1}{2} \cdot (\cot (x) + \tan (x)) = \csc (2 x)$

Step 1

Simplify the left side.

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Step 1.1

Simplify $\frac{1}{2} \cdot (\cot (x) + \tan (x))$ .

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Step 1.1.1

Simplify each term.

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Step 1.1.1.1

Rewrite $\cot (x)$ in terms of sines and cosines.

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

Step 1.1.1.2

Rewrite $\tan (x)$ in terms of sines and cosines.

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

Step 1.1.2

Simplify terms.

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Step 1.1.2.1

Apply the distributive property.

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

Step 1.1.2.2

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

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

Step 1.1.2.3

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

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

Step 2

Simplify the right side.

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Step 2.1

Rewrite $\csc (2 x)$ in terms of sines and cosines.

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

Step 3

Multiply both sides of the equation by $\sin (2 x)$ .

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

Step 4

Apply the distributive property.

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

Step 5

Combine $\sin (2 x)$ and $\frac{\cos (x)}{2 \sin (x)}$ .

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

Step 6

Combine $\sin (2 x)$ and $\frac{\sin (x)}{2 \cos (x)}$ .

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

Step 7

Simplify each term.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Apply the sine double-angle identity.

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

Step 7.1.2

Combine exponents.

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Step 7.1.2.1

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

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

Step 7.1.2.2

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

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

Step 7.1.2.3

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

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

Step 7.1.2.4

Add $1$ and $1$ .

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

Step 7.2

Cancel the common factor of $2$ .

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Step 7.2.1

Cancel the common factor.

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

Step 7.2.2

Rewrite the expression.

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

Step 7.3

Cancel the common factor of $\sin (x)$ .

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Step 7.3.1

Cancel the common factor.

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

Step 7.3.2

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

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

Step 7.4

Simplify the numerator.

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Step 7.4.1

Apply the sine double-angle identity.

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

Step 7.4.2

Combine exponents.

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Step 7.4.2.1

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

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

Step 7.4.2.2

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

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

Step 7.4.2.3

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

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

Step 7.4.2.4

Add $1$ and $1$ .

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

Step 7.5

Cancel the common factor of $2$ .

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Step 7.5.1

Cancel the common factor.

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

Step 7.5.2

Rewrite the expression.

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

Step 7.6

Cancel the common factor of $\cos (x)$ .

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Step 7.6.1

Cancel the common factor.

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

Step 7.6.2

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

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

Step 8

Rearrange terms.

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

Step 9

Apply pythagorean identity.

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

Step 10

Cancel the common factor of $\sin (2 x)$ .

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Step 10.1

Cancel the common factor.

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

Step 10.2

Rewrite the expression.

$1 = 1$

Step 11

Since $1 = 1$ , the equation will always be true for any value of $x$ .

All real numbers

Step 12

The result can be shown in multiple forms.

All real numbers

Interval Notation:

$(- \infty, \infty)$