Precalculus Examples

$\tan (u) (\csc (u) - \sin (u)) = \cos (u)$

Step 1

Start on the left side.

$\tan (u) (\csc (u) - \sin (u))$

Step 2

Simplify the expression.

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Step 2.1

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

$\frac{\sin (u)}{\cos (u)} (\csc (u) - \sin (u))$

Step 2.2

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

$\frac{\sin (u)}{\cos (u)} (\frac{1}{\sin (u)} - \sin (u))$

Step 2.3

Apply the distributive property.

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

Step 2.4

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

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Step 2.4.1

Cancel the common factor.

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

Step 2.4.2

Rewrite the expression.

$\frac{1}{\cos (u)} + \frac{\sin (u)}{\cos (u)} (- \sin (u))$

Step 2.5

Rewrite using the commutative property of multiplication.

$\frac{1}{\cos (u)} - \frac{\sin (u)}{\cos (u)} \sin (u)$

Step 2.6

Multiply $- \frac{\sin (u)}{\cos (u)} \sin (u)$ .

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Step 2.6.1

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

$\frac{1}{\cos (u)} - \frac{\sin (u) \sin (u)}{\cos (u)}$

Step 2.6.2

Raise $\sin (u)$ to the power of $1$ .

$\frac{1}{\cos (u)} - \frac{\sin^{1} (u) \sin (u)}{\cos (u)}$

Step 2.6.3

Raise $\sin (u)$ to the power of $1$ .

$\frac{1}{\cos (u)} - \frac{\sin^{1} (u) \sin^{1} (u)}{\cos (u)}$

Step 2.6.4

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

$\frac{1}{\cos (u)} - \frac{{\sin (u)}^{1 + 1}}{\cos (u)}$

Step 2.6.5

Add $1$ and $1$ .

$\frac{1}{\cos (u)} - \frac{\sin^{2} (u)}{\cos (u)}$

Step 2.7

Combine the numerators over the common denominator.

$\frac{1 - \sin^{2} (u)}{\cos (u)}$

Step 2.8

Apply pythagorean identity.

$\frac{\cos^{2} (u)}{\cos (u)}$

Step 2.9

Cancel the common factor of $\cos^{2} (u)$ and $\cos (u)$ .

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Step 2.9.1

Factor $\cos (u)$ out of $\cos^{2} (u)$ .

$\frac{\cos (u) \cos (u)}{\cos (u)}$

Step 2.9.2

Cancel the common factors.

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Step 2.9.2.1

Multiply by $1$ .

$\frac{\cos (u) \cos (u)}{\cos (u) \cdot 1}$

Step 2.9.2.2

Cancel the common factor.

$\frac{\cos (u) \cos (u)}{\cos (u) \cdot 1}$

Step 2.9.2.3

Rewrite the expression.

$\frac{\cos (u)}{1}$

Step 2.9.2.4

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

$\cos (u)$

Step 3

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

$\tan (u) (\csc (u) - \sin (u)) = \cos (u)$ is an identity