Final answer:
The algebraic expression for sec(cos^-1(u)) is 1/u, utilizing the relationship that secant is the reciprocal of cosine and the Pythagorean identity.
Step-by-step explanation:
The student is asking to express the trigonometric function sec(cos^-1(u)) as an algebraic expression in terms of u. The function cos^-1(u) represents the angle whose cosine is u, and sec is the reciprocal of cosine.
To find the algebraic expression, we can use the Pythagorean identity which states that sin^2(θ) + cos^2(θ) = 1, where θ is an angle. Since u is the cosine of the angle, if cos(θ) = u, then sin(θ) = √(1 - u^2). The secant of the angle is then the reciprocal of u, so sec(θ) = 1/cos(θ) = 1/u. Therefore, the algebraic expression for sec(cos^-1(u)) is 1/u.
The trigonometric expression sec(cos^-1u) can be expressed as an algebraic expression in u using the inverse trigonometric function definition.
Since the inverse cosine function can be expressed as cos^-1(u) = arccos(u), we can rewrite the expression as sec(arccos(u))
Using the definition of secant function, sec(x) = 1/cos(x), the expression can be further simplified as 1/cos(arccos(u)).
Since arccos(u) returns an angle whose cosine is u, the expression can be written as 1/u.