Question

What is tan(sin^-1(x/2))

with steps pls

Answer 1

`(x√(4-x^2))/(4-x^2)`

Explanation:

Given:

`\tan \left(\sin^(-1)\left((x)/(2)\right)\right)`

`\boxed{\begin{minipage}{5cm}\underline{Trigonometric Identity}\\\\$\tan(\arcsin(x))=(x)/(√(1-x^2))$\\ \end{minipage}}`

Use the tan(arcsin(x)) trigonometric identity and replace x for (x/2):

`\implies \tan\left(\arcsin \left((x)/(2)\right)\right)=\frac{\left((x)/(2)\right)}{\sqrt{1-\left((x)/(2)\right)^2}}`

Simplify the denominator:

`\implies \tan\left(\arcsin \left((x)/(2)\right)\right)=\frac{\left((x)/(2)\right)}{\sqrt{1-(x^2)/(4)}}`

`\implies \tan\left(\arcsin \left((x)/(2)\right)\right)=\frac{\left((x)/(2)\right)}{\sqrt{(4-x^2)/(4)}}`

`\implies \tan\left(\arcsin \left((x)/(2)\right)\right)=(\left((x)/(2)\right))/((√(4-x^2))/(√(4)))}`

`\implies \tan\left(\arcsin \left((x)/(2)\right)\right)=(\left((x)/(2)\right))/((√(4-x^2))/(2))}`

`\textsf{Apply the fraction rule}: \quad ((a)/(c))/((b)/(c))=(a)/(b)`

`\implies \tan\left(\arcsin \left((x)/(2)\right)\right)=(x)/(√(4-x^2))`

Multiply the numerator and denominator by the denominator to eliminate the radical from the denominator:

`\implies \tan\left(\arcsin \left((x)/(2)\right)\right)=(x)/(√(4-x^2)) \cdot (√(4-x^2))/(√(4-x^2))`

Simplify:

`\implies \tan\left(\arcsin \left((x)/(2)\right)\right)=(x√(4-x^2))/(4-x^2)`