To write [tex]\(\cos(\arcsin(3x) + \arccos(x))\)[/tex] as an algebraic expression of [tex]\(x\)[/tex] that does not involve trigonometric functions, we can follow these steps:
1. Recall Trigonometric Identities:
[tex]\[
\cos(A + B) = \cos A \cos B - \sin A \sin B
\][/tex]
2. Set Relevant Values:
Let [tex]\(A = \arcsin(3x)\)[/tex] and [tex]\(B = \arccos(x)\)[/tex].
3. Determine [tex]\(\cos A\)[/tex] and [tex]\(\sin A\)[/tex]:
[tex]\[
A = \arcsin(3x) \implies \sin A = 3x \text{ and thus } \cos A = \sqrt{1 - (3x)^2}= \sqrt{1 - 9x^2}
\][/tex]
4. Determine [tex]\(\cos B\)[/tex] and [tex]\(\sin B\)[/tex]:
[tex]\[
B = \arccos(x) \implies \cos B = x \text{ and thus } \sin B = \sqrt{1 - x^2}
\][/tex]
5. Substitute into the Identity:
[tex]\[
\cos(A + B) = \cos (\arcsin(3x) + \arccos(x)) = \cos(\arcsin(3x)) \cos(\arccos(x)) - \sin(\arcsin(3x)) \sin(\arccos(x))
\][/tex]
Substituting the values we know:
[tex]\[
\cos(A + B) = (\sqrt{1 - 9x^2}) x - (3x) \sqrt{1 - x^2}
\][/tex]
6. Simplify the Expression:
[tex]\[
\cos(\arcsin(3x) + \arccos(x)) = x \sqrt{1 - 9x^2} - 3x \sqrt{1 - x^2}
\][/tex]
Thus, the final simplified expression for [tex]\(\cos(\arcsin(3x) + \arccos(x))\)[/tex] in terms of [tex]\(x\)[/tex] is:
[tex]\[
\boxed{x \sqrt{1 - 9x^2} - 3x \sqrt{1 - x^2}}
\][/tex]
