Quadratic Equation:
The given equation is √(1 - sin(a)) = cktan(a).

Derivation:
To solve this equation, we need to manipulate it algebraically to find the value of 'a'. Let's follow the steps below:

Step 1: Simplify the equation:
Square both sides of the equation to eliminate the square root:
1 - sin(a) = (cktan(a))^2

Step 2: Simplify the equation further:
Expand the square on the right side:
1 - sin(a) = c^2k^2tan^2(a)

Step 3: Rewrite tan(a) in terms of sin and cos:
Using the identity tan(a) = sin(a)/cos(a), we can rewrite the equation as:
1 - sin(a) = c^2k^2(sin^2(a)/cos^2(a))

Step 4: Simplify the equation even further:
Multiply both sides by cos^2(a) to get rid of the denominator:
cos^2(a) - sin(a)cos^2(a) = c^2k^2sin^2(a)

Step 5: Rearrange the equation:
Rearrange the terms on the left side of the equation:
cos^2(a) - c^2k^2sin^2(a) - sin(a)cos^2(a) = 0

Step 6: Factorize the equation:
Factor out cos^2(a) from the first two terms and sin(a) from the last two terms:
cos^2(a)(1 - c^2k^2sin^2(a)) - sin(a)cos^2(a) = 0

Step 7: Simplify the equation:
Combine like terms:
cos^2(a)(1 - c^2k^2sin^2(a) - sin(a)) = 0

Step 8: Solve for 'a' in each case:
We have two cases to consider:
1. cos^2(a) = 0
2. 1 - c^2k^2sin^2(a) - sin(a) = 0

Case 1: cos^2(a) = 0:
If cos^2(a) = 0, then cos(a) = 0. This implies that 'a' lies in the quadrants where cos(a) equals zero, i.e., in the second and third quadrants.

Case 2: 1 - c^2k^2sin^2(a) - sin(a) = 0:
To solve this equation, we need to use numerical methods or approximation techniques since it does not have a simple algebraic solution.