Algebra Examples

$\frac{\cos (u) \sec (u)}{\tan (u)} = \frac{f (u)}{g (u)}$

Step 1

Multiply both sides by $\tan (u)$ .

$\frac{\cos (u) \sec (u)}{\tan (u)} \tan (u) = \frac{f u}{g u} \tan (u)$

Step 2

Simplify.

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Step 2.1

Simplify the left side.

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Step 2.1.1

Simplify $\frac{\cos (u) \sec (u)}{\tan (u)} \tan (u)$ .

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Step 2.1.1.1

Cancel the common factor of $\tan (u)$ .

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Step 2.1.1.1.1

Cancel the common factor.

$\frac{\cos (u) \sec (u)}{\tan (u)} \tan (u) = \frac{f u}{g u} \tan (u)$

Step 2.1.1.1.2

Rewrite the expression.

$\cos (u) \sec (u) = \frac{f u}{g u} \tan (u)$

Step 2.1.1.2

Rewrite in terms of sines and cosines, then cancel the common factors.

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Step 2.1.1.2.1

Reorder $\cos (u)$ and $\sec (u)$ .

$\sec (u) \cos (u) = \frac{f u}{g u} \tan (u)$

Step 2.1.1.2.2

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

$\frac{1}{\cos (u)} \cos (u) = \frac{f u}{g u} \tan (u)$

Step 2.1.1.2.3

Cancel the common factors.

$1 = \frac{f u}{g u} \tan (u)$

Step 2.2

Simplify the right side.

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Step 2.2.1

Simplify $\frac{f u}{g u} \tan (u)$ .

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Step 2.2.1.1

Cancel the common factor of $u$ .

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Step 2.2.1.1.1

Cancel the common factor.

$1 = \frac{f u}{g u} \tan (u)$

Step 2.2.1.1.2

Rewrite the expression.

$1 = \frac{f}{g} \tan (u)$

Step 2.2.1.2

Combine $\frac{f}{g}$ and $\tan (u)$ .

$1 = \frac{f \tan (u)}{g}$

Step 3

Solve for $u$ .

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Step 3.1

Rewrite the equation as $\frac{f \tan (u)}{g} = 1$ .

$\frac{f \tan (u)}{g} = 1$

Step 3.2

Multiply both sides by $g$ .

$\frac{f \tan (u)}{g} g = 1 g$

Step 3.3

Simplify.

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Step 3.3.1

Simplify the left side.

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Step 3.3.1.1

Cancel the common factor of $g$ .

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Step 3.3.1.1.1

Cancel the common factor.

$\frac{f \tan (u)}{g} g = 1 g$

Step 3.3.1.1.2

Rewrite the expression.

$f \tan (u) = 1 g$

Step 3.3.2

Simplify the right side.

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Step 3.3.2.1

Multiply $g$ by $1$ .

$f \tan (u) = g$

Step 3.4

Divide each term in $f \tan (u) = g$ by $f$ and simplify.

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Step 3.4.1

Divide each term in $f \tan (u) = g$ by $f$ .

$\frac{f \tan (u)}{f} = \frac{g}{f}$

Step 3.4.2

Simplify the left side.

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Step 3.4.2.1

Cancel the common factor of $f$ .

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Step 3.4.2.1.1

Cancel the common factor.

$\frac{f \tan (u)}{f} = \frac{g}{f}$

Step 3.4.2.1.2

Divide $\tan (u)$ by $1$ .

$\tan (u) = \frac{g}{f}$

Step 3.5

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

$u = \arctan (\frac{g}{f})$