Problem 3 (3 pts) : Among the following two algorithms, which is the best for evaluating f (x) = tan (x) - sin (x) for x ∼ 0? Briefly explain. (a) (1 / cos (x) - 1) sin (x), (b) tan (x) sin2 (x) / (cos (x) + 1).

A) (1 / cos (x) - 1) sin (x)

Step-by-step explanation:

Given the function

f (x) = tan (x) - sin (x)

According to the trigonometry identity, tan (x) = sin (x) / cos (x)

Substituting this into the original equation, we will have;

f (x) = sin (x) / cos (x) - sin (x)

Since sin (x) is common at the numerator, we will factor it out to have;

f (x) = sin (x) {1/cos (x) - 1}

Therefore the first option (1 / cos (x) - 1) sin (x) is the best algorithm for evaluating the function since we could generate the function (1 / cos (x) - 1) sin (x) using the function f (x) = tan (x) - sin (x).