Simplify sec(arctan(( square root of 3)/3))

Problem

sec(arctan(√(,3)/3))

Solution

1. Evaluate the inner expression arctan(√(,3)/3) by identifying the angle θ in the interval (−π/2,π/2) such that tan(θ)=√(,3)/3

2. Recognize that √(,3)/3 is equivalent to 1/√(,3) which corresponds to the tangent of 30 or π/6 radians.

3. Substitute the angle back into the outer function to get sec(π/6)

4. Calculate the secant value using the identity sec(θ)=1/cos(θ)

5. Evaluate cos(π/6)=√(,3)/2 so sec(π/6)=2/√(,3)

6. Rationalize the denominator by multiplying the numerator and denominator by √(,3)

Final Answer

sec(arctan(√(,3)/3))=(2√(,3))/3

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