Expand the Trigonometric Expression tan(arcsin(x+1/2))

$\tan (\arcsin (x + \frac{1}{2}))$

Step 1

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

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

Step 2

Simplify the numerator.

To write $x$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

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

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

Combine the numerators over the common denominator.

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

Move $2$ to the left of $x$ .

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

Step 3

Simplify the denominator.

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

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

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 $b = x + \frac{1}{2}$ .

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

Simplify.

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

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

Combine the numerators over the common denominator.

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

Add $2$ and $1$ .

$\frac{\frac{2 x + 1}{2}}{\sqrt{(x + \frac{3}{2}) (1 - (x + \frac{1}{2}))}}$

To write $x$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{\frac{2 x + 1}{2}}{\sqrt{(x \cdot \frac{2}{2} + \frac{3}{2}) (1 - (x + \frac{1}{2}))}}$

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

$\frac{\frac{2 x + 1}{2}}{\sqrt{(\frac{x \cdot 2}{2} + \frac{3}{2}) (1 - (x + \frac{1}{2}))}}$

Combine the numerators over the common denominator.

$\frac{\frac{2 x + 1}{2}}{\sqrt{\frac{x \cdot 2 + 3}{2} (1 - (x + \frac{1}{2}))}}$

Move $2$ to the left of $x$ .

$\frac{\frac{2 x + 1}{2}}{\sqrt{\frac{2 x + 3}{2} (1 - (x + \frac{1}{2}))}}$

To write $x$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{\frac{2 x + 1}{2}}{\sqrt{\frac{2 x + 3}{2} (1 - (x \cdot \frac{2}{2} + \frac{1}{2}))}}$

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

$\frac{\frac{2 x + 1}{2}}{\sqrt{\frac{2 x + 3}{2} (1 - (\frac{x \cdot 2}{2} + \frac{1}{2}))}}$

Combine the numerators over the common denominator.

$\frac{\frac{2 x + 1}{2}}{\sqrt{\frac{2 x + 3}{2} (1 - \frac{x \cdot 2 + 1}{2})}}$

Move $2$ to the left of $x$ .

$\frac{\frac{2 x + 1}{2}}{\sqrt{\frac{2 x + 3}{2} (1 - \frac{2 x + 1}{2})}}$

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

$\frac{\frac{2 x + 1}{2}}{\sqrt{\frac{2 x + 3}{2} (\frac{2}{2} - \frac{2 x + 1}{2})}}$

Combine the numerators over the common denominator.

$\frac{\frac{2 x + 1}{2}}{\sqrt{\frac{2 x + 3}{2} \cdot \frac{2 - (2 x + 1)}{2}}}$

Rewrite $\frac{2 - (2 x + 1)}{2}$ in a factored form.

Apply the distributive property.

$\frac{\frac{2 x + 1}{2}}{\sqrt{\frac{2 x + 3}{2} \cdot \frac{2 - (2 x) - 1 \cdot 1}{2}}}$

Multiply $2$ by $- 1$ .

$\frac{\frac{2 x + 1}{2}}{\sqrt{\frac{2 x + 3}{2} \cdot \frac{2 - 2 x - 1 \cdot 1}{2}}}$

Multiply $- 1$ by $1$ .

$\frac{\frac{2 x + 1}{2}}{\sqrt{\frac{2 x + 3}{2} \cdot \frac{2 - 2 x - 1}{2}}}$

Subtract $1$ from $2$ .

$\frac{\frac{2 x + 1}{2}}{\sqrt{\frac{2 x + 3}{2} \cdot \frac{- 2 x + 1}{2}}}$

Multiply $\frac{2 x + 3}{2}$ by $\frac{- 2 x + 1}{2}$ .

$\frac{\frac{2 x + 1}{2}}{\sqrt{\frac{(2 x + 3) (- 2 x + 1)}{2 \cdot 2}}}$

Multiply $2$ by $2$ .

$\frac{\frac{2 x + 1}{2}}{\sqrt{\frac{(2 x + 3) (- 2 x + 1)}{4}}}$

Rewrite $\frac{(2 x + 3) (- 2 x + 1)}{4}$ as ${(\frac{1}{2})}^{2} ((2 x + 3) (- 2 x + 1))$ .

Factor the perfect power $1^{2}$ out of $(2 x + 3) (- 2 x + 1)$ .

$\frac{\frac{2 x + 1}{2}}{\sqrt{\frac{1^{2} ((2 x + 3) (- 2 x + 1))}{4}}}$

Factor the perfect power $2^{2}$ out of $4$ .

$\frac{\frac{2 x + 1}{2}}{\sqrt{\frac{1^{2} ((2 x + 3) (- 2 x + 1))}{2^{2} \cdot 1}}}$

Rearrange the fraction $\frac{1^{2} ((2 x + 3) (- 2 x + 1))}{2^{2} \cdot 1}$ .

$\frac{\frac{2 x + 1}{2}}{\sqrt{{(\frac{1}{2})}^{2} ((2 x + 3) (- 2 x + 1))}}$

Pull terms out from under the radical.

$\frac{\frac{2 x + 1}{2}}{\frac{1}{2} \sqrt{(2 x + 3) (- 2 x + 1)}}$

Combine $\frac{1}{2}$ and $\sqrt{(2 x + 3) (- 2 x + 1)}$ .

$\frac{\frac{2 x + 1}{2}}{\frac{\sqrt{(2 x + 3) (- 2 x + 1)}}{2}}$

Step 4

Multiply the numerator by the reciprocal of the denominator.

$\frac{2 x + 1}{2} \cdot \frac{2}{\sqrt{(2 x + 3) (- 2 x + 1)}}$

Step 5

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x + 1}{2} \cdot \frac{2}{\sqrt{(2 x + 3) (- 2 x + 1)}}$

Rewrite the expression.

$(2 x + 1) \frac{1}{\sqrt{(2 x + 3) (- 2 x + 1)}}$

Step 6

Multiply $\frac{1}{\sqrt{(2 x + 3) (- 2 x + 1)}}$ by $\frac{\sqrt{(2 x + 3) (- 2 x + 1)}}{\sqrt{(2 x + 3) (- 2 x + 1)}}$ .

$(2 x + 1) (\frac{1}{\sqrt{(2 x + 3) (- 2 x + 1)}} \cdot \frac{\sqrt{(2 x + 3) (- 2 x + 1)}}{\sqrt{(2 x + 3) (- 2 x + 1)}})$

Step 7

Combine and simplify the denominator.

Multiply $\frac{1}{\sqrt{(2 x + 3) (- 2 x + 1)}}$ by $\frac{\sqrt{(2 x + 3) (- 2 x + 1)}}{\sqrt{(2 x + 3) (- 2 x + 1)}}$ .

$(2 x + 1) \frac{\sqrt{(2 x + 3) (- 2 x + 1)}}{\sqrt{(2 x + 3) (- 2 x + 1)} \sqrt{(2 x + 3) (- 2 x + 1)}}$

Raise $\sqrt{(2 x + 3) (- 2 x + 1)}$ to the power of $1$ .

$(2 x + 1) \frac{\sqrt{(2 x + 3) (- 2 x + 1)}}{{\sqrt{(2 x + 3) (- 2 x + 1)}}^{1} \sqrt{(2 x + 3) (- 2 x + 1)}}$

Raise $\sqrt{(2 x + 3) (- 2 x + 1)}$ to the power of $1$ .

$(2 x + 1) \frac{\sqrt{(2 x + 3) (- 2 x + 1)}}{{\sqrt{(2 x + 3) (- 2 x + 1)}}^{1} {\sqrt{(2 x + 3) (- 2 x + 1)}}^{1}}$

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

$(2 x + 1) \frac{\sqrt{(2 x + 3) (- 2 x + 1)}}{{\sqrt{(2 x + 3) (- 2 x + 1)}}^{1 + 1}}$

Add $1$ and $1$ .

$(2 x + 1) \frac{\sqrt{(2 x + 3) (- 2 x + 1)}}{{\sqrt{(2 x + 3) (- 2 x + 1)}}^{2}}$

Rewrite ${\sqrt{(2 x + 3) (- 2 x + 1)}}^{2}$ as $(2 x + 3) (- 2 x + 1)$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{(2 x + 3) (- 2 x + 1)}$ as ${((2 x + 3) (- 2 x + 1))}^{\frac{1}{2}}$ .

$(2 x + 1) \frac{\sqrt{(2 x + 3) (- 2 x + 1)}}{{({((2 x + 3) (- 2 x + 1))}^{\frac{1}{2}})}^{2}}$

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

$(2 x + 1) \frac{\sqrt{(2 x + 3) (- 2 x + 1)}}{{((2 x + 3) (- 2 x + 1))}^{\frac{1}{2} \cdot 2}}$

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

$(2 x + 1) \frac{\sqrt{(2 x + 3) (- 2 x + 1)}}{{((2 x + 3) (- 2 x + 1))}^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$(2 x + 1) \frac{\sqrt{(2 x + 3) (- 2 x + 1)}}{{((2 x + 3) (- 2 x + 1))}^{\frac{2}{2}}}$

Rewrite the expression.

$(2 x + 1) \frac{\sqrt{(2 x + 3) (- 2 x + 1)}}{{((2 x + 3) (- 2 x + 1))}^{1}}$

Simplify.

$(2 x + 1) \frac{\sqrt{(2 x + 3) (- 2 x + 1)}}{(2 x + 3) (- 2 x + 1)}$

Step 8

Multiply $2 x + 1$ by $\frac{\sqrt{(2 x + 3) (- 2 x + 1)}}{(2 x + 3) (- 2 x + 1)}$ .

$\frac{(2 x + 1) \sqrt{(2 x + 3) (- 2 x + 1)}}{(2 x + 3) (- 2 x + 1)}$

Step 9

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

$\frac{(2 x + 1) \sqrt{(2 x + 3) (- 2 x + 1)}}{(2 x + 3) (- (2 x) + 1)}$

Step 10

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

$\frac{(2 x + 1) \sqrt{(2 x + 3) (- 2 x + 1)}}{(2 x + 3) (- (2 x) - 1 (- 1))}$

Step 11

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

$\frac{(2 x + 1) \sqrt{(2 x + 3) (- 2 x + 1)}}{(2 x + 3) (- (2 x - 1))}$

Step 12

Rewrite negatives.

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

$\frac{(2 x + 1) \sqrt{(2 x + 3) (- 2 x + 1)}}{(2 x + 3) (- 1 (2 x - 1))}$

Move the negative in front of the fraction.

$- \frac{(2 x + 1) \sqrt{(2 x + 3) (- 2 x + 1)}}{(2 x + 3) (2 x - 1)}$

Reorder factors in $- \frac{(2 x + 1) \sqrt{(2 x + 3) (- 2 x + 1)}}{(2 x + 3) (2 x - 1)}$ .

$- \frac{\sqrt{(2 x + 3) (- 2 x + 1)} (2 x + 1)}{(2 x + 3) (2 x - 1)}$

Expand the Trigonometric Expression tan(arcsin(x+1/2))

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