Final answer:
To solve Sin(x) - Cos(x) = 0 and find the value of sec(x), we can rewrite the equation as Sin(x) = Cos(x) and then substitute Cos^2(x) with 1 - Sin^2(x). Solving this quadratic equation will give us the values of Sin(x) and hence sec(x). The possible values of sec(x) are √5 ± 1.
Step-by-step explanation:
To solve the equation Sin(x) - Cos(x) = 0, we can rewrite it as Sin(x) = Cos(x). We know that Sin^2(x) + Cos^2(x) = 1 (from the trigonometric identity). So, we can substitute Cos^2(x) in the equation with 1 - Sin^2(x) to get Sin(x) = 1 - Sin^2(x). Rearranging the equation, we get Sin^2(x) + Sin(x) - 1 = 0. Solving this quadratic equation will give us the values of Sin(x) and hence the value of sec(x).
Using the quadratic formula, we have Sin(x) = (-1 ± √5) / 2. Since sec(x) = 1 / Cos(x), we can find the value of sec(x) by using the reciprocal of the value of Cos(x). Therefore, the possible values of sec(x) are 1 / Cos(x) = 1 / (√5 ± 1). Simplifying, we get sec(x) = √5 ± 1.