Trigonometry Examples

$\frac{\sin (x + h) - \sin (x)}{h} = \cos (x) (\frac{\sin (h)}{h}) - \sin (x) \frac{1 - \cos (h)}{h}$

Step 1

Start on the left side.

$\frac{\sin (x + h) - \sin (x)}{h}$

Step 2

Apply the sum of angles identity.

$\frac{\sin (x) \cos (h) + \cos (x) \sin (h) - \sin (x)}{h}$

Step 3

Simplify.

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Step 3.1

Factor $- 1$ out of $- \sin (x)$ .

$\frac{- (\sin (x)) + \cos (h) \sin (x) + \cos (x) \sin (h)}{h}$

Step 3.2

Factor $- 1$ out of $\cos (h) \sin (x)$ .

$\frac{- (\sin (x)) - (- \cos (h) \sin (x)) + \cos (x) \sin (h)}{h}$

Step 3.3

Factor $- 1$ out of $- (\sin (x)) - (- \cos (h) \sin (x))$ .

$\frac{- (\sin (x) - \cos (h) \sin (x)) + \cos (x) \sin (h)}{h}$

Step 3.4

Factor $- 1$ out of $\cos (x) \sin (h)$ .

$\frac{- (\sin (x) - \cos (h) \sin (x)) - (- \cos (x) \sin (h))}{h}$

Step 3.5

Factor $- 1$ out of $- (\sin (x) - \cos (h) \sin (x)) - (- \cos (x) \sin (h))$ .

$\frac{- (\sin (x) - \cos (h) \sin (x) - \cos (x) \sin (h))}{h}$

Step 3.6

Rewrite $- (\sin (x) - \cos (h) \sin (x) - \cos (x) \sin (h))$ as $- 1 (\sin (x) - \cos (h) \sin (x) - \cos (x) \sin (h))$ .

$\frac{- 1 (\sin (x) - \cos (h) \sin (x) - \cos (x) \sin (h))}{h}$

Step 3.7

Move the negative in front of the fraction.

$- \frac{\sin (x) - \cos (h) \sin (x) - \cos (x) \sin (h)}{h}$

Step 4

Now consider the right side of the equation.

$\cos (x) (\frac{\sin (h)}{h}) - \sin (x) \frac{1 - \cos (h)}{h}$

Step 5

Simplify each term.

$\frac{\cos (x) \sin (h)}{h} - \frac{\sin (x)}{h} + \frac{\sin (x) \cos (h)}{h}$

Step 6

Write $\frac{\cos (x) \sin (h)}{h} - \frac{\sin (x)}{h}$ as a fraction with denominator $1$ .

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

Step 7

Add fractions.

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Step 7.1

To write $\frac{\frac{\cos (x) \sin (h)}{h} - \frac{\sin (x)}{h}}{1}$ as a fraction with a common denominator, multiply by $\frac{h}{h}$ .

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

Step 7.2

Multiply $\frac{\frac{\cos (x) \sin (h)}{h} - \frac{\sin (x)}{h}}{1}$ by $\frac{h}{h}$ .

$\frac{(\frac{\cos (x) \sin (h)}{h} - \frac{\sin (x)}{h}) h}{h} + \frac{\sin (x) \cos (h)}{h}$

Step 7.3

Combine the numerators over the common denominator.

$\frac{(\frac{\cos (x) \sin (h)}{h} - \frac{\sin (x)}{h}) h + \sin (x) \cos (h)}{h}$

Step 8

Simplify each term.

$\frac{\cos (x) \sin (h) - \sin (x) + \sin (x) \cos (h)}{h}$

Step 9

Simplify.

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Step 9.1

Factor $- 1$ out of $- \sin (x)$ .

$\frac{- (\sin (x)) + \cos (h) \sin (x) + \cos (x) \sin (h)}{h}$

Step 9.2

Factor $- 1$ out of $\cos (h) \sin (x)$ .

$\frac{- (\sin (x)) - (- \cos (h) \sin (x)) + \cos (x) \sin (h)}{h}$

Step 9.3

Factor $- 1$ out of $- (\sin (x)) - (- \cos (h) \sin (x))$ .

$\frac{- (\sin (x) - \cos (h) \sin (x)) + \cos (x) \sin (h)}{h}$

Step 9.4

Factor $- 1$ out of $\cos (x) \sin (h)$ .

$\frac{- (\sin (x) - \cos (h) \sin (x)) - (- \cos (x) \sin (h))}{h}$

Step 9.5

Factor $- 1$ out of $- (\sin (x) - \cos (h) \sin (x)) - (- \cos (x) \sin (h))$ .

$\frac{- (\sin (x) - \cos (h) \sin (x) - \cos (x) \sin (h))}{h}$

Step 9.6

Rewrite $- (\sin (x) - \cos (h) \sin (x) - \cos (x) \sin (h))$ as $- 1 (\sin (x) - \cos (h) \sin (x) - \cos (x) \sin (h))$ .

$\frac{- 1 (\sin (x) - \cos (h) \sin (x) - \cos (x) \sin (h))}{h}$

Step 9.7

Move the negative in front of the fraction.

$- \frac{\sin (x) - \cos (h) \sin (x) - \cos (x) \sin (h)}{h}$

Step 10

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

$\frac{\sin (x + h) - \sin (x)}{h} = \cos (x) (\frac{\sin (h)}{h}) - \sin (x) \frac{1 - \cos (h)}{h}$ is an identity