Precalculus Examples

$\frac{\sin (y) + \tan (y)}{1 + \sec (y)} = \sin (y)$

Step 1

Start on the left side.

$\frac{\sin (y) + \tan (y)}{1 + \sec (y)}$

Step 2

Convert to sines and cosines.

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Step 2.1

Write $\tan (y)$ in sines and cosines using the quotient identity.

$\frac{\sin (y) + \frac{\sin (y)}{\cos (y)}}{1 + \sec (y)}$

Step 2.2

Apply the reciprocal identity to $\sec (y)$ .

$\frac{\sin (y) + \frac{\sin (y)}{\cos (y)}}{1 + \frac{1}{\cos (y)}}$

Step 3

Simplify.

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Step 3.1

Multiply the numerator and denominator of the fraction by $\cos (y)$ .

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Step 3.1.1

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

$\frac{\cos (y)}{\cos (y)} \cdot \frac{\sin (y) + \frac{\sin (y)}{\cos (y)}}{1 + \frac{1}{\cos (y)}}$

Step 3.1.2

Combine.

$\frac{\cos (y) (\sin (y) + \frac{\sin (y)}{\cos (y)})}{\cos (y) (1 + \frac{1}{\cos (y)})}$

Step 3.2

Apply the distributive property.

$\frac{\cos (y) \sin (y) + \cos (y) \frac{\sin (y)}{\cos (y)}}{\cos (y) \cdot 1 + \cos (y) \frac{1}{\cos (y)}}$

Step 3.3

Simplify by cancelling.

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Step 3.3.1

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

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Step 3.3.1.1

Cancel the common factor.

$\frac{\cos (y) \sin (y) + \cos (y) \frac{\sin (y)}{\cos (y)}}{\cos (y) \cdot 1 + \cos (y) \frac{1}{\cos (y)}}$

Step 3.3.1.2

Rewrite the expression.

$\frac{\cos (y) \sin (y) + \sin (y)}{\cos (y) \cdot 1 + \cos (y) \frac{1}{\cos (y)}}$

Step 3.3.2

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

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Step 3.3.2.1

Cancel the common factor.

$\frac{\cos (y) \sin (y) + \sin (y)}{\cos (y) \cdot 1 + \cos (y) \frac{1}{\cos (y)}}$

Step 3.3.2.2

Rewrite the expression.

$\frac{\cos (y) \sin (y) + \sin (y)}{\cos (y) \cdot 1 + 1}$

Step 3.4

Factor $\sin (y)$ out of $\cos (y) \sin (y) + \sin (y)$ .

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Step 3.4.1

Multiply by $1$ .

$\frac{\sin (y) \cos (y) + \sin (y) \cdot 1}{\cos (y) \cdot 1 + 1}$

Step 3.4.2

Factor $\sin (y)$ out of $\sin (y) \cos (y) + \sin (y) \cdot 1$ .

$\frac{\sin (y) (\cos (y) + 1)}{\cos (y) \cdot 1 + 1}$

Step 3.5

Multiply $\cos (y)$ by $1$ .

$\frac{\sin (y) (\cos (y) + 1)}{\cos (y) + 1}$

Step 3.6

Cancel the common factor of $\cos (y) + 1$ .

$\sin (y)$

Step 4

Because the two sides have been shown to be equivalent, the equation is an identity.

$\frac{\sin (y) + \tan (y)}{1 + \sec (y)} = \sin (y)$ is an identity