Simplify sin(arcsin(x)+arccos(x))

Problem

sin(arcsin(x)+arccos(x))

Solution

1. Identify the trigonometric identity for the sum of two angles, which states sin(α+β)=sin(α)*cos(β)+cos(α)*sin(β)

2. Assign the variables α=arcsin(x) and β=arccos(x)

3. Apply the definitions of inverse trigonometric functions, noting that sin(arcsin(x))=x and cos(arccos(x))=x for x in the domain [−1,1]

4. Determine the remaining terms using the Pythagorean identity cos(arcsin(x))=√(,1−x2) and sin(arccos(x))=√(,1−x2)

5. Substitute these values into the sum identity:

sin(arcsin(x)+arccos(x))=(x)*(x)+(√(,1−x2))*(√(,1−x2))

6. Simplify the resulting expression:

x2+(1−x2)=1

Final Answer

sin(arcsin(x)+arccos(x))=1

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