Simplify sin(arctan(2x)-arccos(x))

Problem

sin(arctan(2*x)−arccos(x))

Solution

1. Identify the expression as a sine of a difference of two angles, sin(α−β) where α=arctan(2*x) and β=arccos(x)

2. Apply the identity for the sine of a difference:

sin(α−β)=sin(α)*cos(β)−cos(α)*sin(β)

3. Determine the trigonometric values for α=arctan(2*x) Let α be an angle in a right triangle where the opposite side is 2*x and the adjacent side is 1 The hypotenuse is √(,(2*x)2+1)=√(,4*x2+1)

sin(arctan(2*x))=(2*x)/√(,4*x2+1)

cos(arctan(2*x))=1/√(,4*x2+1)

4. Determine the trigonometric values for β=arccos(x) Let β be an angle in a right triangle where the adjacent side is x and the hypotenuse is 1 The opposite side is √(,1−x2)=√(,1−x2)

cos(arccos(x))=x

sin(arccos(x))=√(,1−x2)

5. Substitute these values back into the difference identity:

sin(α−β)=((2*x)/√(,4*x2+1))*(x)−(1/√(,4*x2+1))*(√(,1−x2))

6. Simplify the expression by combining the terms over a common denominator:

sin(α−β)=(2*x2−√(,1−x2))/√(,4*x2+1)

Final Answer

sin(arctan(2*x)−arccos(x))=(2*x2−√(,1−x2))/√(,4*x2+1)

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