Trigonometry Examples

$\sec (2 x) = \frac{\sec^{2} (x)}{2 - \sec^{2} (x)}$

Step 1

Multiply both sides by $2 - \sec^{2} (x)$ .

$\sec (2 x) (2 - \sec^{2} (x)) = \frac{\sec^{2} (x)}{2 - \sec^{2} (x)} (2 - \sec^{2} (x))$

Step 2

Simplify.

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Step 2.1

Simplify the left side.

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Step 2.1.1

Simplify $\sec (2 x) (2 - \sec^{2} (x))$ .

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Step 2.1.1.1

Apply the distributive property.

$\sec (2 x) \cdot 2 + \sec (2 x) (- \sec^{2} (x)) = \frac{\sec^{2} (x)}{2 - \sec^{2} (x)} (2 - \sec^{2} (x))$

Step 2.1.1.2

Reorder.

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Step 2.1.1.2.1

Move $2$ to the left of $\sec (2 x)$ .

$2 \cdot \sec (2 x) + \sec (2 x) (- \sec^{2} (x)) = \frac{\sec^{2} (x)}{2 - \sec^{2} (x)} (2 - \sec^{2} (x))$

Step 2.1.1.2.2

Rewrite using the commutative property of multiplication.

$2 \sec (2 x) - \sec (2 x) \sec^{2} (x) = \frac{\sec^{2} (x)}{2 - \sec^{2} (x)} (2 - \sec^{2} (x))$

Step 2.2

Simplify the right side.

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Step 2.2.1

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

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Step 2.2.1.1

Cancel the common factor.

$2 \sec (2 x) - \sec (2 x) \sec^{2} (x) = \frac{\sec^{2} (x)}{2 - \sec^{2} (x)} (2 - \sec^{2} (x))$

Step 2.2.1.2

Rewrite the expression.

$2 \sec (2 x) - \sec (2 x) \sec^{2} (x) = \sec^{2} (x)$

Step 3

Solve for $x$ .

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Step 3.1

Simplify the left side.

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Step 3.1.1

Simplify each term.

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Step 3.1.1.1

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

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

Step 3.1.1.2

Combine $2$ and $\frac{1}{\cos (2 x)}$ .

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

Step 3.1.1.3

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

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

Step 3.1.1.4

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

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

Step 3.1.1.5

Apply the product rule to $\frac{1}{\cos (x)}$ .

$\frac{2}{\cos (2 x)} - \frac{1}{\cos (2 x)} \cdot \frac{1^{2}}{\cos^{2} (x)} = \sec^{2} (x)$

Step 3.1.1.6

One to any power is one.

$\frac{2}{\cos (2 x)} - \frac{1}{\cos (2 x)} \cdot \frac{1}{\cos^{2} (x)} = \sec^{2} (x)$

Step 3.1.1.7

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

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

Step 3.2

Simplify the right side.

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Step 3.2.1

Simplify $\sec^{2} (x)$ .

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Step 3.2.1.1

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

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

Step 3.2.1.2

Apply the product rule to $\frac{1}{\cos (x)}$ .

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

Step 3.2.1.3

One to any power is one.

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

Step 3.3

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

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

Step 3.4

Apply the distributive property.

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

Step 3.5

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

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Step 3.5.1

Cancel the common factor.

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

Step 3.5.2

Rewrite the expression.

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

Step 3.6

Rewrite using the commutative property of multiplication.

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

Step 3.7

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

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Step 3.7.1

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

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

Step 3.7.2

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

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

Step 3.7.3

Cancel the common factor.

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

Step 3.7.4

Rewrite the expression.

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

Step 3.8

Combine $\cos (2 x)$ and $\frac{1}{\cos^{2} (x)}$ .

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

Step 3.9

Multiply both sides by $\cos^{2} (x)$ .

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

Step 3.10

Simplify.

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Step 3.10.1

Simplify the left side.

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Step 3.10.1.1

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

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Step 3.10.1.1.1

Move the leading negative in $- \frac{1}{\cos^{2} (x)}$ into the numerator.

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

Step 3.10.1.1.2

Cancel the common factor.

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

Step 3.10.1.1.3

Rewrite the expression.

$- 1 = (\sec^{2} (x) \cos (2 x) - 2) \cos^{2} (x)$

Step 3.10.2

Simplify the right side.

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Step 3.10.2.1

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

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Step 3.10.2.1.1

Simplify each term.

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Step 3.10.2.1.1.1

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

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

Step 3.10.2.1.1.2

Apply the product rule to $\frac{1}{\cos (x)}$ .

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

Step 3.10.2.1.1.3

One to any power is one.

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

Step 3.10.2.1.1.4

Combine $\frac{1}{\cos^{2} (x)}$ and $\cos (2 x)$ .

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

Step 3.10.2.1.2

Simplify terms.

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Step 3.10.2.1.2.1

Apply the distributive property.

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

Step 3.10.2.1.2.2

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

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Step 3.10.2.1.2.2.1

Cancel the common factor.

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

Step 3.10.2.1.2.2.2

Rewrite the expression.

$- 1 = \cos (2 x) - 2 \cos^{2} (x)$

Step 3.11

Solve for $x$ .

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Step 3.11.1

Rewrite the equation as $\cos (2 x) - 2 \cos^{2} (x) = - 1$ .

$\cos (2 x) - 2 \cos^{2} (x) = - 1$

Step 3.11.2

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

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

Step 3.11.3

Subtract $1$ from both sides of the equation.

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

Step 3.11.4

Simplify the left side.

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Step 3.11.4.1

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

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Step 3.11.4.1.1

Simplify with factoring out.

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Step 3.11.4.1.1.1

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

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

Step 3.11.4.1.1.2

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

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

Step 3.11.4.1.1.3

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

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

Step 3.11.4.1.2

Apply pythagorean identity.

$- 2 \cdot 1 = - 1 - 1$

Step 3.11.4.1.3

Multiply $- 2$ by $1$ .

$- 2 = - 1 - 1$

Step 3.11.5

Simplify the right side.

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Step 3.11.5.1

Subtract $1$ from $- 1$ .

$- 2 = - 2$

Step 3.11.6

Since $- 2 = - 2$ , the equation will always be true for any value of $x$ .

All real numbers

Step 4

The result can be shown in multiple forms.

All real numbers

Interval Notation:

$(- \infty, \infty)$