Precalculus Examples

$\frac{\sin (t)}{\sin (t) + \cos (t)} = \frac{\tan (t)}{A + \tan (t)}$

Step 1

Rewrite the equation as $\frac{\tan (t)}{A + \tan (t)} = \frac{\sin (t)}{\sin (t) + \cos (t)}$ .

$\frac{\tan (t)}{A + \tan (t)} = \frac{\sin (t)}{\sin (t) + \cos (t)}$

Step 2

Multiply the numerator of the first fraction by the denominator of the second fraction. Set this equal to the product of the denominator of the first fraction and the numerator of the second fraction.

$\tan (t) (\sin (t) + \cos (t)) = (A + \tan (t)) \sin (t)$

Step 3

Solve the equation for $A$ .

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Step 3.1

Rewrite the equation as $(A + \tan (t)) \sin (t) = \tan (t) (\sin (t) + \cos (t))$ .

$(A + \tan (t)) \sin (t) = \tan (t) (\sin (t) + \cos (t))$

Step 3.2

Divide each term in $(A + \tan (t)) \sin (t) = \tan (t) (\sin (t) + \cos (t))$ by $\sin (t)$ and simplify.

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Step 3.2.1

Divide each term in $(A + \tan (t)) \sin (t) = \tan (t) (\sin (t) + \cos (t))$ by $\sin (t)$ .

$\frac{(A + \tan (t)) \sin (t)}{\sin (t)} = \frac{\tan (t) (\sin (t) + \cos (t))}{\sin (t)}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $\sin (t)$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{(A + \tan (t)) \sin (t)}{\sin (t)} = \frac{\tan (t) (\sin (t) + \cos (t))}{\sin (t)}$

Step 3.2.2.1.2

Divide $A + \tan (t)$ by $1$ .

$A + \tan (t) = \frac{\tan (t) (\sin (t) + \cos (t))}{\sin (t)}$

Step 3.2.2.2

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

$A + \frac{\sin (t)}{\cos (t)} = \frac{\tan (t) (\sin (t) + \cos (t))}{\sin (t)}$

Step 3.2.2.3

Convert from $\frac{\sin (t)}{\cos (t)}$ to $\tan (t)$ .

$A + \tan (t) = \frac{\tan (t) (\sin (t) + \cos (t))}{\sin (t)}$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Separate fractions.

$A + \tan (t) = \frac{\sin (t) + \cos (t)}{1} \cdot \frac{\tan (t)}{\sin (t)}$

Step 3.2.3.2

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

$A + \tan (t) = \frac{\sin (t) + \cos (t)}{1} \cdot \frac{\frac{\sin (t)}{\cos (t)}}{\sin (t)}$

Step 3.2.3.3

Rewrite $\frac{\frac{\sin (t)}{\cos (t)}}{\sin (t)}$ as a product.

$A + \tan (t) = \frac{\sin (t) + \cos (t)}{1} (\frac{\sin (t)}{\cos (t)} \cdot \frac{1}{\sin (t)})$

Step 3.2.3.4

Cancel the common factor of $\sin (t)$ .

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Step 3.2.3.4.1

Cancel the common factor.

$A + \tan (t) = \frac{\sin (t) + \cos (t)}{1} (\frac{\sin (t)}{\cos (t)} \cdot \frac{1}{\sin (t)})$

Step 3.2.3.4.2

Rewrite the expression.

$A + \tan (t) = \frac{\sin (t) + \cos (t)}{1} \cdot \frac{1}{\cos (t)}$

Step 3.2.3.5

Divide $\sin (t) + \cos (t)$ by $1$ .

$A + \tan (t) = (\sin (t) + \cos (t)) \frac{1}{\cos (t)}$

Step 3.2.3.6

Apply the distributive property.

$A + \tan (t) = \sin (t) \frac{1}{\cos (t)} + \cos (t) \frac{1}{\cos (t)}$

Step 3.2.3.7

Combine $\sin (t)$ and $\frac{1}{\cos (t)}$ .

$A + \tan (t) = \frac{\sin (t)}{\cos (t)} + \cos (t) \frac{1}{\cos (t)}$

Step 3.2.3.8

Cancel the common factor of $\cos (t)$ .

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Step 3.2.3.8.1

Cancel the common factor.

$A + \tan (t) = \frac{\sin (t)}{\cos (t)} + \cos (t) \frac{1}{\cos (t)}$

Step 3.2.3.8.2

Rewrite the expression.

$A + \tan (t) = \frac{\sin (t)}{\cos (t)} + 1$

Step 3.2.3.9

Convert from $\frac{\sin (t)}{\cos (t)}$ to $\tan (t)$ .

$A + \tan (t) = \tan (t) + 1$

Step 3.3

Move all terms not containing $A$ to the right side of the equation.

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Step 3.3.1

Subtract $\tan (t)$ from both sides of the equation.

$A = \tan (t) + 1 - \tan (t)$

Step 3.3.2

Combine the opposite terms in $\tan (t) + 1 - \tan (t)$ .

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Step 3.3.2.1

Subtract $\tan (t)$ from $\tan (t)$ .

$A = 0 + 1$

Step 3.3.2.2

Add $0$ and $1$ .

$A = 1$