Mathematics: Find the exact value of the following expression (without using a calculator): tan(sin^-1 x/2)

Find the exact value of the following expression

kalbaugh

10.01.2022

tan(Sin^-1 x/2)= $\frac{x/2}{\sqrt{1-x^{2}/4 } }$

Step-by-step explanation:

Let sin^-1 x/2= θ

then sinθ= x/2

on the basis of unit circle, we have a triangle with hypotenuse of length 1, one side of length x/2 and opposite angle of θ.

tan(Sin^-1 x/2) = tanθ

tanθ= sinθ/cosθ

as per trigonometric identities cosθ= √(1-sin^2θ)

tanθ= sinθ/ √(1-sin^2θ)

substituting the value sinθ=x/2 in the above equation

tanθ= $\frac{x/2}{\sqrt{1-x^{2}/4 } }$

now substituting the value sin^-1 x/2= θ in above equation

tan(sin^-1 x/2) = $\frac{x/2}{\sqrt{1-x^{2}/4 } }$

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