# How do you simplify Cos(arccos(2x) + arcsin(x))?

Jul 24, 2016

cos 3x

#### Explanation:

arccos 2x --> 2x
arcsin x --> sin x
cos (arccos 2x + arcsin x) = cos (2x + x) = cos 3x

Jul 24, 2016

$\pm 2 x \sqrt{1 - {x}^{2}} \pm x \sqrt{1 - 4 {x}^{2}} , - \frac{1}{2} \le x \le \frac{1}{2}$

#### Explanation:

arc cos 2x is the angle whose cosine is 2x.

Let $a = a r c \cos \left(2 x\right)$.

Then, $\cos a = 2 x \in \left[- 1 , 1\right] \mathmr{and} \sin a = \pm \sqrt{1 - 2 {x}^{2}}$

Note that $x \in \left[- \frac{1}{2.} \frac{1}{2}\right]$.

Prefix negative sign for principal value $a > \frac{\pi}{4}$ (when x > 0)..

Let $b = a r c \sin x$.

Then, $\sin b = x \mathmr{and} \cos b = \pm \sqrt{1 - {x}^{2}}$.

Prefix negative sign for principal value $b < 0$ (when x < 0).

Now, the given expression is

$\cos \left(a + b\right)$

$= \cos a \cos b - \sin a \sin b$

$= \pm 2 x \sqrt{1 - {x}^{2}} \pm x \sqrt{1 - 4 {x}^{2}}$