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

Question

Quizplus · Accepted Answer

To find the exact value of $\cos(\tan^{-1}x)$, we can use a right triangle where the opposite side over the adjacent side (the definition of tangent) equals $x$. Let's consider a right triangle where the opposite side is $x$ and the adjacent side is $1$, making the hypotenuse $\sqrt{x^2 + 1}$ by the Pythagorean theorem. The cosine of an angle is the adjacent side over the hypotenuse, so $\cos(\tan^{-1}x) = \frac{1}{\sqrt{x^2 + 1}}$, which simplifies to $\frac{\sqrt{x^2 + 1}}{x^2 + 1}$ when rationalized.

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

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