Algebra Examples

$r = \tan (θ) \sec (θ)$

Step 1

Simplify the right side.

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Step 1.1

Simplify $\tan (θ) \sec (θ)$ .

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Step 1.1.1

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

$r = \frac{\sin (θ)}{\cos (θ)} \sec (θ)$

Step 1.1.2

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

$r = \frac{\sin (θ)}{\cos (θ)} \cdot \frac{1}{\cos (θ)}$

Step 1.1.3

Multiply $\frac{\sin (θ)}{\cos (θ)} \cdot \frac{1}{\cos (θ)}$ .

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Step 1.1.3.1

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

$r = \frac{\sin (θ)}{\cos (θ) \cos (θ)}$

Step 1.1.3.2

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

$r = \frac{\sin (θ)}{\cos^{1} (θ) \cos (θ)}$

Step 1.1.3.3

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

$r = \frac{\sin (θ)}{\cos^{1} (θ) \cos^{1} (θ)}$

Step 1.1.3.4

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

$r = \frac{\sin (θ)}{{\cos (θ)}^{1 + 1}}$

Step 1.1.3.5

Add $1$ and $1$ .

$r = \frac{\sin (θ)}{\cos^{2} (θ)}$

Step 2

Since $\sin (θ) = \frac{y}{r}$ , replace $\sin (θ)$ with $\frac{y}{r}$ .

$r = \frac{\frac{y}{r}}{\cos^{2} (θ)}$

Step 3

Since $\cos (θ) = \frac{x}{r}$ , replace $\cos (θ)$ with $\frac{x}{r}$ .

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

Step 4

Multiply both sides by $r$ .

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Step 4.1

Multiply both sides by $r$ .

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

Step 4.2

Simplify the left side.

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Step 4.2.1

Multiply $r$ by $r$ .

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

Step 4.3

Simplify the right side.

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Step 4.3.1

Simplify $r \frac{\frac{y}{r}}{{(\frac{x}{r})}^{2}}$ .

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Step 4.3.1.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 4.3.1.2

Combine.

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

Step 4.3.1.3

Cancel the common factor of $r$ .

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Step 4.3.1.3.1

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

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

Step 4.3.1.3.2

Cancel the common factor.

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

Step 4.3.1.3.3

Rewrite the expression.

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

Step 4.3.1.4

Simplify the expression.

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Step 4.3.1.4.1

Multiply $y$ by $1$ .

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

Step 4.3.1.4.2

Apply the product rule to $\frac{x}{r}$ .

$r^{2} = \frac{y}{\frac{x^{2}}{r^{2}}}$

Step 4.3.1.5

Multiply the numerator by the reciprocal of the denominator.

$r^{2} = y \frac{r^{2}}{x^{2}}$

Step 4.3.1.6

Combine $y$ and $\frac{r^{2}}{x^{2}}$ .

$r^{2} = \frac{y r^{2}}{x^{2}}$

Step 5

Since $r^{2} = x^{2} + y^{2}$ , replace $r^{2}$ with $x^{2} + y^{2}$ and $r$ with $\sqrt{x^{2} + y^{2}}$ .

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