Trigonometry Examples

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

Step 1

Start on the right side.

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

Step 2

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

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

Step 3

Combine.

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

Step 4

Simplify numerator.

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Step 4.1

Apply the distributive property.

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

Step 4.2

Multiply $\sin (u)$ by $1$ .

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

Step 4.3

Reorder factors in $\sin (u) + \sin (u) (- \cos (u))$ .

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

Step 5

Simplify denominator.

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Step 5.1

Expand $(1 + \cos (u)) (1 - \cos (u))$ using the FOIL Method.

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Step 5.1.1

Apply the distributive property.

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

Step 5.1.2

Apply the distributive property.

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

Step 5.1.3

Apply the distributive property.

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

Step 5.2

Simplify and combine like terms.

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

Step 6

Apply pythagorean identity.

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

Step 7

Simplify.

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Step 7.1

Factor $\sin (u)$ out of $\sin (u) - \sin (u) \cos (u)$ .

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Step 7.1.1

Multiply by $1$ .

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

Step 7.1.2

Factor $\sin (u)$ out of $- \sin (u) \cos (u)$ .

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

Step 7.1.3

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

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

Step 7.2

Cancel the common factors.

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

Step 8

Now consider the left side of the equation.

$\csc (u) - \cot (u)$

Step 9

Convert to sines and cosines.

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Step 9.1

Apply the reciprocal identity to $\csc (u)$ .

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

Step 9.2

Write $\cot (u)$ in sines and cosines using the quotient identity.

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

Step 10

Combine the numerators over the common denominator.

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

Step 11

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

$\csc (u) - \cot (u) = \frac{\sin (u)}{1 + \cos (u)}$ is an identity