Find the exact value of the expression, if possible. Do not use a calculator.
-$\sin \left( \tan ^ { - 1 } x \right)$
A) $\frac { x \sqrt { x ^ { 2 } + 1 } } { x ^ { 2 } + 1 }$
B) $\frac { x \sqrt { x ^ { 2 } - 1 } } { x ^ { 2 } - 1 }$
C) $x \sqrt { x ^ { 2 } + 1 }$
D) $\frac { \sqrt { x ^ { 2 } + 1 } } { x ^ { 2 } + 1 }$

Question

Quizplus · Accepted Answer

To find the exact value of $\sin(\tan^{-1}x)$, consider a right triangle where the angle $\theta = \tan^{-1}x$. This means that the opposite side over the adjacent side (the definition of tangent) is $x$. If we let the opposite side be $x$ and the adjacent side be $1$, then by the Pythagorean theorem, the hypotenuse is $\sqrt{x^2 + 1}$. Therefore, $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{x}{\sqrt{x^2 + 1}}$, which simplifies to choice A.

Find the Exact Value of the Expression, If Possible sin⁡(tan⁡−1x)\sin \left( \tan ^ { - 1 } x \right)sin(tan−1x)

Find the Exact Value of the Expression, If Possible $\sin \left( \tan ^ { - 1 } x \right)$