Precalculus Examples

$x (t) = \tan (t)$ , $y (t) = \sec^{2} (t)$

Step 1

Set up the parametric equation for $x (t)$ to solve the equation for $t$ .

$x = \tan (t)$

Step 2

Rewrite the equation as $\tan (t) = x$ .

$\tan (t) = x$

Step 3

Take the inverse tangent of both sides of the equation to extract $t$ from inside the tangent.

$t = \arctan (x)$

Step 4

Replace $t$ in the equation for $y$ to get the equation in terms of $x$ .

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

Step 5

Remove parentheses.

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

Step 6

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

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Step 6.1

Draw a triangle in the plane with vertices $(1, x)$ , $(1, 0)$ , and the origin. Then $\arctan (x)$ is the angle between the positive x-axis and the ray beginning at the origin and passing through $(1, x)$ . Therefore, $\sec (\arctan (x))$ is $\sqrt{1 + x^{2}}$ .

$y = {\sqrt{1 + x^{2}}}^{2}$

Step 6.2

Rewrite ${\sqrt{1 + x^{2}}}^{2}$ as $1 + x^{2}$ .

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Step 6.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{1 + x^{2}}$ as ${(1 + x^{2})}^{\frac{1}{2}}$ .

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

Step 6.2.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 6.2.3

Combine $\frac{1}{2}$ and $2$ .

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

Step 6.2.4

Cancel the common factor of $2$ .

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Step 6.2.4.1

Cancel the common factor.

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

Step 6.2.4.2

Rewrite the expression.

$y = {(1 + x^{2})}^{1}$

Step 6.2.5

Simplify.

$y = 1 + x^{2}$

Step 6.3

Reorder $1$ and $x^{2}$ .

$y = x^{2} + 1$