Simplify sin(2arctan(x))

Problem

sin(2*arctan(x))

Solution

1. Identify the double angle identity for sine, which states sin(2*θ)=2*sin(θ)*cos(θ)

2. Substitute θ=arctan(x) into the identity to rewrite the expression.

sin(2*arctan(x))=2*sin(arctan(x))*cos(arctan(x))

3. Construct a right triangle where θ=arctan(x) In this triangle, the side opposite to θ is x and the adjacent side is 1

4. Determine the hypotenuse using the Pythagorean theorem, which results in √(,x2+1)=√(,x2+1)

5. Evaluate the trigonometric functions sin(θ) and cos(θ) based on the triangle ratios.

sin(arctan(x))=x/√(,x2+1)

cos(arctan(x))=1/√(,x2+1)

6. Substitute these ratios back into the double angle formula.

sin(2*arctan(x))=2⋅x/√(,x2+1)⋅1/√(,x2+1)

7. Simplify the expression by multiplying the fractions.

sin(2*arctan(x))=(2*x)/(x2+1)

Final Answer

sin(2*arctan(x))=(2*x)/(x2+1)

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