Solve for x sec(x)^4-tan(x)^4=1+2tan(x)^2

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Trigonometry

$\sec^{4} (x) - \tan^{4} (x) = 1 + 2 \tan^{2} (x)$

Simplify the left side.

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Simplify the expression.

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Rewrite $\sec^{4} (x)$ as ${(\sec^{2} (x))}^{2}$ .

${(\sec^{2} (x))}^{2} - \tan^{4} (x) = 1 + 2 \tan^{2} (x)$

Rewrite $\tan^{4} (x)$ as ${(\tan^{2} (x))}^{2}$ .

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

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = \sec^{2} (x)$ and $b = \tan^{2} (x)$ .

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

Apply pythagorean identity.

$(\sec^{2} (x) + \tan^{2} (x)) \cdot 1 = 1 + 2 \tan^{2} (x)$

Multiply $\sec^{2} (x) + \tan^{2} (x)$ by $1$ .

$\sec^{2} (x) + \tan^{2} (x) = 1 + 2 \tan^{2} (x)$

Move all terms not containing $\sec (x)$ to the right side of the equation.

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Subtract $\tan^{2} (x)$ from both sides of the equation.

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

Subtract $\tan^{2} (x)$ from $2 \tan^{2} (x)$ .

$\sec^{2} (x) = 1 + \tan^{2} (x)$

Take the $square$ root of both sides of the $equation$ to eliminate the exponent on the left side.

$\sec (x) = \pm \sqrt{1 + \tan^{2} (x)}$

The complete solution is the result of both the positive and negative portions of the solution.

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First, use the positive value of the $\pm$ to find the first solution.

$\sec (x) = \sqrt{1 + \tan^{2} (x)}$

Next, use the negative value of the $\pm$ to find the second solution.

$\sec (x) = - \sqrt{1 + \tan^{2} (x)}$

The complete solution is the result of both the positive and negative portions of the solution.

$\sec (x) = \sqrt{1 + \tan^{2} (x)}, - \sqrt{1 + \tan^{2} (x)}$

Set up each of the solutions to solve for $x$ .

$\sec (x) = \sqrt{1 + \tan^{2} (x)}$

$\sec (x) = - \sqrt{1 + \tan^{2} (x)}$

Set up the equation to solve for $x$ .

$\sec (x) = \sqrt{1 + \tan^{2} (x)}$

Solve the equation for $x$ .

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Since the radical is on the right side of the equation, switch the sides so it is on the left side of the equation.

$\sqrt{1 + \tan^{2} (x)} = \sec (x)$

To remove the radical on the left side of the equation, square both sides of the equation.

${(\sqrt{1 + \tan^{2} (x)})}^{2} = \sec^{2} (x)$

Simplify the left side of the equation.

$1 + \tan^{2} (x) = \sec^{2} (x)$

Solve for $x$ .

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Replace the $\tan^{2} (x)$ with $\sec^{2} (x) - 1$ based on the $\tan^{2} (x) + 1 = \sec^{2} (x)$ identity.

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

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

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Subtract $1$ from $1$ .

$\sec^{2} (x) + 0 = \sec^{2} (x)$

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

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

For the two functions to be equal, the arguments of each must be equal.

$x = x$

Move all terms containing $x$ to the left side of the equation.

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Subtract $x$ from both sides of the equation.

$x - x = 0$

Subtract $x$ from $x$ .

$0 = 0$

Since $0 = 0$ , the equation will always be true.

All real numbers

Set up the equation to solve for $x$ .

$\sec (x) = - \sqrt{1 + \tan^{2} (x)}$

Solve the equation for $x$ .

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Since the radical is on the right side of the equation, switch the sides so it is on the left side of the equation.

$- \sqrt{1 + \tan^{2} (x)} = \sec (x)$

To remove the radical on the left side of the equation, square both sides of the equation.

${(- \sqrt{1 + \tan^{2} (x)})}^{2} = \sec^{2} (x)$

Simplify the left side of the equation.

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

Solve for $x$ .

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Simplify the left side.

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Rearrange terms.

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

Apply pythagorean identity.

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

Simplify the expression.

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Raise $- 1$ to the power of $2$ .

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

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

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

For the two functions to be equal, the arguments of each must be equal.

$x = x$

Move all terms containing $x$ to the left side of the equation.

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Subtract $x$ from both sides of the equation.

$x - x = 0$

Subtract $x$ from $x$ .

$0 = 0$

Since $0 = 0$ , the equation will always be true.

All real numbers

The complete solution is the set of all solutions.

No solution

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