Trigonometry Examples

$\tan (\arcsin (x - 1))$

Step 1

Draw a triangle in the plane with vertices $(\sqrt{1^{2} - {(x - 1)}^{2}}, x - 1)$ , $(\sqrt{1^{2} - {(x - 1)}^{2}}, 0)$ , and the origin. Then $\arcsin (x - 1)$ is the angle between the positive x-axis and the ray beginning at the origin and passing through $(\sqrt{1^{2} - {(x - 1)}^{2}}, x - 1)$ . Therefore, $\tan (\arcsin (x - 1))$ is $\frac{x - 1}{\sqrt{1 - {(x - 1)}^{2}}}$ .

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

Step 2

Simplify the denominator.

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Step 2.1

Rewrite $1$ as $1^{2}$ .

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

Step 2.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 1$ and $b = x - 1$ .

$\frac{x - 1}{\sqrt{(1 + x - 1) (1 - (x - 1))}}$

Step 2.3

Simplify.

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Step 2.3.1

Subtract $1$ from $1$ .

$\frac{x - 1}{\sqrt{(x + 0) (1 - (x - 1))}}$

Step 2.3.2

Add $x$ and $0$ .

$\frac{x - 1}{\sqrt{x (1 - (x - 1))}}$

Step 2.3.3

Apply the distributive property.

$\frac{x - 1}{\sqrt{x (1 - x - - 1)}}$

Step 2.3.4

Multiply $- 1$ by $- 1$ .

$\frac{x - 1}{\sqrt{x (1 - x + 1)}}$

Step 2.3.5

Add $1$ and $1$ .

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

Step 3

Multiply $\frac{x - 1}{\sqrt{x (- x + 2)}}$ by $\frac{\sqrt{x (- x + 2)}}{\sqrt{x (- x + 2)}}$ .

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

Step 4

Combine and simplify the denominator.

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Step 4.1

Multiply $\frac{x - 1}{\sqrt{x (- x + 2)}}$ by $\frac{\sqrt{x (- x + 2)}}{\sqrt{x (- x + 2)}}$ .

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

Step 4.2

Raise $\sqrt{x (- x + 2)}$ to the power of $1$ .

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

Step 4.3

Raise $\sqrt{x (- x + 2)}$ to the power of $1$ .

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

Step 4.4

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

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

Step 4.5

Add $1$ and $1$ .

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

Step 4.6

Rewrite ${\sqrt{x (- x + 2)}}^{2}$ as $x (- x + 2)$ .

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Step 4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x (- x + 2)}$ as ${(x (- x + 2))}^{\frac{1}{2}}$ .

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

Step 4.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 4.6.3

Combine $\frac{1}{2}$ and $2$ .

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

Step 4.6.4

Cancel the common factor of $2$ .

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Step 4.6.4.1

Cancel the common factor.

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

Step 4.6.4.2

Rewrite the expression.

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

Step 4.6.5

Simplify.

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

Step 5

Factor $- 1$ out of $- x$ .

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

Step 6

Rewrite $2$ as $- 1 (- 2)$ .

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

Step 7

Factor $- 1$ out of $- (x) - 1 (- 2)$ .

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

Step 8

Rewrite negatives.

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Step 8.1

Rewrite $- (x - 2)$ as $- 1 (x - 2)$ .

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

Step 8.2

Move the negative in front of the fraction.

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

Step 8.3

Reorder factors in $- \frac{(x - 1) \sqrt{x (- x + 2)}}{x (x - 2)}$ .

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