2 tan2(t) = 3 sec(t) Find all exact solutions on [0, 2π)

Answer 1

2tan^2(x)= 3sec(x) Change tan to sin/cos and sec to 1/cos

2(sin^2(x)/cos^2(x)= 3/cos(x) Cross multiply

2sin^2(x)=3cos(x). Use identity sin^2 = 1-cos ^2

2(1-cos^2(x))=3cos(x)

2 -2cos^2(x)=3cos(x)

2cos^2(x)+3cos(x)-2=0 Factor

(2cosx-1)(cosx+2) =0

cosx= 1/2 This would be at pi/3and 5pi/3

cosx=-2 no solution .There is no angle with a cosine value

Final answer pi/3 and 5pi/3