Trigonometry Examples

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

Step 1

Simplify the left side.

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Step 1.1

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

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Step 1.1.1

Simplify the numerator.

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Step 1.1.1.1

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

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

Step 1.1.1.2

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

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

Step 1.1.2

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

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

Step 1.1.3

Multiply the numerator by the reciprocal of the denominator.

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

Step 1.1.4

Apply the distributive property.

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

Step 1.1.5

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

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Step 1.1.5.1

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

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

Step 1.1.5.2

Raise $\sin (2 x)$ to the power of $1$ .

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

Step 1.1.5.3

Raise $\sin (2 x)$ to the power of $1$ .

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

Step 1.1.5.4

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

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

Step 1.1.5.5

Add $1$ and $1$ .

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

Step 1.1.6

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

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Step 1.1.6.1

Cancel the common factor.

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

Step 1.1.6.2

Rewrite the expression.

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

Step 2

Simplify the right side.

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Step 2.1

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

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

Step 3

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

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

Step 4

Apply the distributive property.

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

Step 5

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

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Step 5.1

Cancel the common factor.

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

Step 5.2

Rewrite the expression.

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

Step 6

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

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Step 6.1

Raise $\cos (2 x)$ to the power of $1$ .

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

Step 6.2

Raise $\cos (2 x)$ to the power of $1$ .

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

Step 6.3

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

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

Step 6.4

Add $1$ and $1$ .

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

Step 7

Apply pythagorean identity.

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

Step 8

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

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Step 8.1

Cancel the common factor.

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

Step 8.2

Rewrite the expression.

$1 = 1$

Step 9

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

All real numbers

Step 10

The result can be shown in multiple forms.

All real numbers

Interval Notation:

$(- \infty, \infty)$