Simplify cos(arcsin(x)-arctan(y))

Problem

cos(arcsin(x)−arctan(y))

Solution

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

2. Apply the formula for the cosine of a difference, which is cos(α−β)=cos(α)*cos(β)+sin(α)*sin(β)

3. Define the trigonometric relationships for α=arcsin(x) This implies sin(α)=x Using the identity cos2(α)+sin2(α)=1 we find cos(α)=√(,1−x2)

4. Define the trigonometric relationships for β=arctan(y) This implies tan(β)=y Visualizing 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 expanded cosine difference formula.

cos(arcsin(x)−arctan(y))=√(,1−x2)⋅1/√(,1+y2)+x⋅y/√(,1+y2)

6. Combine the terms over a common denominator.

cos(arcsin(x)−arctan(y))=(√(,1−x2)+x*y)/√(,1+y2)

Final Answer

cos(arcsin(x)−arctan(y))=(√(,1−x2)+x*y)/√(,1+y2)

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