Evaluate the following expression.
4 tan x (arcsin x)

A) $\frac { - 4 x \sqrt { 1 + x ^ { 2 } } } { 1 + x ^ { 2 } }$
B) $\frac { 4 x \sqrt { 1 + x ^ { 2 } } } { 1 + x ^ { 2 } }$
C) $\frac { - 4 x \sqrt { 1 - x ^ { 2 } } } { 1 - x ^ { 2 } }$
D) $\frac { 4 x \sqrt { x ^ { 2 } - 1 } } { x ^ { 2 } - 1 }$
E) $\frac { 4 x \sqrt { 1 - x ^ { 2 } } } { 1 - x ^ { 2 } }$

Question

Evaluate the following expression.  
4 tan x (arcsin x)
 
A) $\frac { - 4 x \sqrt { 1 + x ^ { 2 } } } { 1 + x ^ { 2 } }$
B) $\frac { 4 x \sqrt { 1 + x ^ { 2 } } } { 1 + x ^ { 2 } }$
C) $\frac { - 4 x \sqrt { 1 - x ^ { 2 } } } { 1 - x ^ { 2 } }$
D) $\frac { 4 x \sqrt { x ^ { 2 } - 1 } } { x ^ { 2 } - 1 }$
E) $\frac { 4 x \sqrt { 1 - x ^ { 2 } } } { 1 - x ^ { 2 } }$

Quizplus · Accepted Answer

The expression involves trigonometric and inverse trigonometric functions. The key is to understand the relationship between the tangent function and the arcsine function. The tangent of an angle in a right triangle is the opposite side over the adjacent side. If we consider a triangle where the angle is arcsin(x), then the opposite side is x and the hypotenuse is 1. Using the Pythagorean theorem, the adjacent side (which we can call y) would be $\sqrt{1 - x^2}$. Thus, $\tan(\arcsin(x)) = \frac{x}{\sqrt{1 - x^2}}$. Multiplying this by 4 gives the expression $\frac{4x}{\sqrt{1 - x^2}}$, which matches the structure of option E.

Evaluate the Following Expression −4x1+x21+x2\frac { - 4 x \sqrt { 1 + x ^ { 2 } } } { 1 + x ^ { 2 } }1+x2−4x1+x2​​

Evaluate the Following Expression $\frac { - 4 x \sqrt { 1 + x ^ { 2 } } } { 1 + x ^ { 2 } }$