Simplify tan(arcsin(( square root of 2)/2))

$\tan (\arcsin (\frac{\sqrt{2}}{2}))$

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

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

Multiply the numerator by the reciprocal of the denominator.

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

Simplify the denominator.

Tap for more steps...

One to any power is one.

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

Apply the product rule to $\frac{\sqrt{2}}{2}$ .

$\frac{\sqrt{2}}{2} \cdot \frac{1}{\sqrt{1 - \frac{{\sqrt{2}}^{2}}{2^{2}}}}$

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

Tap for more steps...

Rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

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

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

$\frac{\sqrt{2}}{2} \cdot \frac{1}{\sqrt{1 - \frac{2^{\frac{1}{2} \cdot 2}}{2^{2}}}}$

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

$\frac{\sqrt{2}}{2} \cdot \frac{1}{\sqrt{1 - \frac{2^{\frac{2}{2}}}{2^{2}}}}$

Cancel the common factor of $2$ .

Tap for more steps...

Cancel the common factor.

$\frac{\sqrt{2}}{2} \cdot \frac{1}{\sqrt{1 - \frac{2^{\frac{2}{2}}}{2^{2}}}}$

Divide $1$ by $1$ .

$\frac{\sqrt{2}}{2} \cdot \frac{1}{\sqrt{1 - \frac{2^{1}}{2^{2}}}}$

Evaluate the exponent.

$\frac{\sqrt{2}}{2} \cdot \frac{1}{\sqrt{1 - \frac{2}{2^{2}}}}$

Raise $2$ to the power of $2$ .

$\frac{\sqrt{2}}{2} \cdot \frac{1}{\sqrt{1 - \frac{2}{4}}}$

Cancel the common factor of $2$ and $4$ .

Tap for more steps...

Factor $2$ out of $2$ .

$\frac{\sqrt{2}}{2} \cdot \frac{1}{\sqrt{1 - \frac{2 (1)}{4}}}$

Cancel the common factors.

Tap for more steps...

Factor $2$ out of $4$ .

$\frac{\sqrt{2}}{2} \cdot \frac{1}{\sqrt{1 - \frac{2 \cdot 1}{2 \cdot 2}}}$

Cancel the common factor.

$\frac{\sqrt{2}}{2} \cdot \frac{1}{\sqrt{1 - \frac{2 \cdot 1}{2 \cdot 2}}}$

Rewrite the expression.

$\frac{\sqrt{2}}{2} \cdot \frac{1}{\sqrt{1 - \frac{1}{2}}}$

To write $\frac{1}{1}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{\sqrt{2}}{2} \cdot \frac{1}{\sqrt{\frac{1}{1} \cdot \frac{2}{2} - \frac{1}{2}}}$

Write each expression with a common denominator of $2$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps...

Combine.

$\frac{\sqrt{2}}{2} \cdot \frac{1}{\sqrt{\frac{1 \cdot 2}{1 \cdot 2} - \frac{1}{2}}}$

Multiply $2$ by $1$ .

$\frac{\sqrt{2}}{2} \cdot \frac{1}{\sqrt{\frac{1 \cdot 2}{2} - \frac{1}{2}}}$

Combine the numerators over the common denominator.

$\frac{\sqrt{2}}{2} \cdot \frac{1}{\sqrt{\frac{1 \cdot 2 - 1}{2}}}$

Simplify the numerator.

Tap for more steps...

Multiply $1$ by $2$ .

$\frac{\sqrt{2}}{2} \cdot \frac{1}{\sqrt{\frac{2 - 1}{2}}}$

Subtract $1$ from $2$ .

$\frac{\sqrt{2}}{2} \cdot \frac{1}{\sqrt{\frac{1}{2}}}$

Rewrite $\sqrt{\frac{1}{2}}$ as $\frac{\sqrt{1}}{\sqrt{2}}$ .

$\frac{\sqrt{2}}{2} \cdot \frac{1}{\frac{\sqrt{1}}{\sqrt{2}}}$

Any root of $1$ is $1$ .

$\frac{\sqrt{2}}{2} \cdot \frac{1}{\frac{1}{\sqrt{2}}}$

Multiply $\frac{1}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{\sqrt{2}}{2} \cdot \frac{1}{\frac{1}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}}$

Combine and simplify the denominator.

Tap for more steps...

Multiply $\frac{1}{\sqrt{2}}$ and $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{\sqrt{2}}{2} \cdot \frac{1}{\frac{\sqrt{2}}{\sqrt{2} \sqrt{2}}}$

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{\sqrt{2}}{2} \cdot \frac{1}{\frac{\sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}}}$

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{\sqrt{2}}{2} \cdot \frac{1}{\frac{\sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}}$

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

$\frac{\sqrt{2}}{2} \cdot \frac{1}{\frac{\sqrt{2}}{{\sqrt{2}}^{1 + 1}}}$

Add $1$ and $1$ .

$\frac{\sqrt{2}}{2} \cdot \frac{1}{\frac{\sqrt{2}}{{\sqrt{2}}^{2}}}$

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

Tap for more steps...

Rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

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

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

$\frac{\sqrt{2}}{2} \cdot \frac{1}{\frac{\sqrt{2}}{2^{\frac{1}{2} \cdot 2}}}$

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

$\frac{\sqrt{2}}{2} \cdot \frac{1}{\frac{\sqrt{2}}{2^{\frac{2}{2}}}}$

Cancel the common factor of $2$ .

Tap for more steps...

Cancel the common factor.

$\frac{\sqrt{2}}{2} \cdot \frac{1}{\frac{\sqrt{2}}{2^{\frac{2}{2}}}}$

Divide $1$ by $1$ .

$\frac{\sqrt{2}}{2} \cdot \frac{1}{\frac{\sqrt{2}}{2^{1}}}$

Evaluate the exponent.

$\frac{\sqrt{2}}{2} \cdot \frac{1}{\frac{\sqrt{2}}{2}}$

Multiply the numerator by the reciprocal of the denominator.

$\frac{\sqrt{2}}{2} (1 \frac{2}{\sqrt{2}})$

Multiply $\frac{2}{\sqrt{2}}$ by $1$ .

$\frac{\sqrt{2}}{2} \cdot \frac{2}{\sqrt{2}}$

Cancel the common factor of $\sqrt{2}$ .

Tap for more steps...

Cancel the common factor.

$\frac{\sqrt{2}}{2} \cdot \frac{2}{\sqrt{2}}$

Rewrite the expression.

$\frac{1}{2} \cdot 2$

Cancel the common factor of $2$ .

Tap for more steps...

Cancel the common factor.

$\frac{1}{2} \cdot 2$

Rewrite the expression.

$1$

Do you know how to Simplify tan(arcsin(( square root of 2)/2))? If not, you can write to our math experts in our application. The best solution for your task you can find above on this page.