I am very confused too, but I'll try my best.
question 1
cos(x) = sqrt(1-sine(x)^2)
cos(x) = sqrt(1 - (-6k/(k^2+9))^2)
cos(x) = sqrt(1-(36k^2)/(k^4+18k^2+81))
cos(x) = sqrt((k^4-18k^2+81)/(k^4+18^2+81))
cos(x) = (k^2-9)/(k^2+9)
question 2
tan(x)+1/cos(x) = tan(x)+1/cos(x)
tan(x)+1/cos(x) = sine(x)/cos(x) + 1/(cos(x))
tan(x)+1/cos(x) = (sine(x)+1)/cos(x)
tan(x)+1/cos(x) = (-6k/(k^2+9)+1)/((k^2-9)/(k^2+9))
tan(x)+1/cos(x) = ((k^2-6k+9)/(k^2+9))/((k^2-9)/(k^2+9))
tan(x)+1/cos(x) = ((k^2-6k+9)*(k^2+9))/((k^2+9)*(k^2-9))
tan(x)+1/cos(x) = (k^2-6k+9)/(k^2-9)
tan(x)+1/cos(x) = (k-3)(k-3)/(k+3)(k-3)
tan(x)+1/cos(x) = (k-3)/(k+3)
I am about 99% that I am wrong, but I hope this helped. :))
=^._.^=