Number of Real Roots of the Equation

Given Equation: secA + cosecA = √15

Analysis:
To determine the number of real roots of the given equation lying between 0 and 2π, we need to first simplify the equation and then analyze its behavior within the given interval.

Simplification:
secA = 1/cosA
cosecA = 1/sinA
Substitute these values in the given equation:
1/cosA + 1/sinA = √15
Multiplying through by cosA * sinA:
sinA + cosA = √15sinAcosA
Using trigonometric identities:
sinA + cosA = 2sin(A + π/4)
Therefore, the equation simplifies to:
2sin(A + π/4) = √15sinAcosA

Analysis within the Interval [0, 2π]:
Within the interval [0, 2π], the equation will have real roots where the LHS and RHS intersect. Since the LHS is a sinusoidal function with an amplitude of 2 and the RHS is a constant value multiplied by sinAcosA, there will be multiple points of intersection.
To find the exact number of real roots, we need to analyze the behavior of both sides of the equation and determine the points of intersection. This can be done by graphing the functions or using numerical methods.
Therefore, the number of real roots of the equation secA + cosecA = √15 lying between 0 and 2π will be multiple, and the exact number can be determined through further analysis or computation.