Final answer:
The trigonometric expression cos(arccos(8x)-arcsin(8x)) can be simplified as an algebraic expression using the Pythagorean identity and the cosine difference identity, resulting in the expression 8x√(1 - (8x)^2).
Step-by-step explanation:
To write the trigonometric expression cos(arccos(8x)−arcsin(8x)) as an algebraic expression, we can use trigonometric identities to simplify the expression.
First, recognize that arccos(8x) and arcsin(8x) refer to the angles whose cosine and sine are 8x, respectively. Since sine and cosine functions are related via the Pythagorean identity: sin2(A) + cos2(A) = 1, where A is any angle, we can infer that if cos(A) = 8x, then sin(A) = √(1 - (8x)2) assuming A is in the first or fourth quadrant where cosine is positive.
Similarly, if arcsin(8x) is the angle whose sine is 8x, implying cos(B) = √(1 - (8x)2) for angle B in the first or second quadrant.
Now, we can express the trigonometric expression as cos(A - B) using the cosine difference identity: cos(A - B) = cos(A)cos(B) + sin(A)sin(B).
Substituting the algebraic expressions for sine and cosine of A and B, we get cos(arccos(8x) - arcsin(8x)) = (8x)√(1 - (8x)2) + √(1 - (8x)2)*(8x).
This simplifies to (8x)2 + (8x)2(1 - (8x)2), which further simplifies to 8x√(1 - (8x)2) as the algebraic expression.