You are given the value of tan Θ. is it possible to find the value of sec Θ without finding the measure of Θ? Explain.?

No, that is not possible. However, it is possible to derive sec²(Θ).

sec²(Θ) = 1 + tan²(Θ)

Deriving sec(Θ) from tan(Θ) alone is not possible, but the answer must be one of these two values:

sec(Θ) = ±√[1 + tan²(Θ)]

The secant and tangent functions have the same domain, so sec(Θ) exists wherever tan(Θ) exists.

No.

Tangent is positive in Quadrants I and III.

Secant is positive in Quadrants I and IV.

tan(30°) = 1/√3

sec(30°) = 2/√3

tan(210°) = 1/√3

sec(210°) = -2/√3

IF you know the quadrant of the angle, you can find the secant knowing the tangent.

If they give you a rational value, you can use soh cah toa ; tan = opposite/adjacent and then Pythagorean

to get the hypotenuse, then use sec = hypotenuse/adjacent (reciprocal of cosine).

Like if tan Θ = 2/5 and Θ is acute, 2 = opposite, 5 = adjacent

so hypotenuse = √(2² + 5²) = √29 making sec Θ = (√29) / 5

sec(t)^2 - tan(t)^2 = 1 for all defined values of t. So unless t = pi/2 + pi * k, where k is an integer, then you can find sec(t), and if t = pi/2 + pi * k, then sec(t) is undefined