Trigonometry Examples

$1 + x + \frac{x^{2}}{1 - x} = \frac{1}{1 - x}$

Step 1

Write $1$ as a fraction with a common denominator.

$x + \frac{1 - x}{1 - x} + \frac{x^{2}}{1 - x} = \frac{1}{1 - x}$

Step 2

Combine the numerators over the common denominator.

$x + \frac{1 - x + x^{2}}{1 - x} = \frac{1}{1 - x}$

Step 3

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

$\frac{x (1 - x)}{1 - x} + \frac{1 - x + x^{2}}{1 - x} = \frac{1}{1 - x}$

Step 4

Combine the numerators over the common denominator.

$\frac{x (1 - x) + 1 - x + x^{2}}{1 - x} = \frac{1}{1 - x}$

Step 5

Simplify the numerator.

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Step 5.1

Apply the distributive property.

$\frac{x \cdot 1 + x (- x) + 1 - x + x^{2}}{1 - x} = \frac{1}{1 - x}$

Step 5.2

Multiply $x$ by $1$ .

$\frac{x + x (- x) + 1 - x + x^{2}}{1 - x} = \frac{1}{1 - x}$

Step 5.3

Rewrite using the commutative property of multiplication.

$\frac{x - x \cdot x + 1 - x + x^{2}}{1 - x} = \frac{1}{1 - x}$

Step 5.4

Multiply $x$ by $x$ by adding the exponents.

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Step 5.4.1

Move $x$ .

$\frac{x - (x \cdot x) + 1 - x + x^{2}}{1 - x} = \frac{1}{1 - x}$

Step 5.4.2

Multiply $x$ by $x$ .

$\frac{x - x^{2} + 1 - x + x^{2}}{1 - x} = \frac{1}{1 - x}$

Step 5.5

Subtract $x$ from $x$ .

$\frac{- x^{2} + 1 + 0 + x^{2}}{1 - x} = \frac{1}{1 - x}$

Step 5.6

Add $- x^{2}$ and $0$ .

$\frac{- x^{2} + 1 + x^{2}}{1 - x} = \frac{1}{1 - x}$

Step 5.7

Add $- x^{2}$ and $x^{2}$ .

$\frac{0 + 1}{1 - x} = \frac{1}{1 - x}$

Step 5.8

Add $0$ and $1$ .

$\frac{1}{1 - x} = \frac{1}{1 - x}$

Step 6

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

$1 + x + \frac{x^{2}}{1 - x} = \frac{1}{1 - x}$ is an identity.