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

Problem

sin(arccos(x)−arctan(y))

Solution

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

2. Apply the angle subtraction formula for sine, which states sin(α−β)=sin(α)*cos(β)−cos(α)*sin(β)

3. Determine the trigonometric values for α=arccos(x) By definition, cos(α)=x Using the identity sin2(α)+cos2(α)=1 we find sin(α)=√(,1−x2)

4. Determine the trigonometric values for β=arctan(y) By definition, tan(β)=y which can be viewed as a right triangle with opposite side y and adjacent side 1 The hypotenuse is √(,1+y2) Thus, sin(β)=y/√(,1+y2) and cos(β)=1/√(,1+y2)

5. Substitute these values back into the subtraction formula:

sin(arccos(x)−arctan(y))=(√(,1−x2))*(1/√(,1+y2))−(x)*(y/√(,1+y2))

6. Combine the terms over the common denominator √(,1+y2)

Final Answer

sin(arccos(x)−arctan(y))=(√(,1−x2)−x*y)/√(,1+y2)

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