Question

i need to solve this over the interval [0,2pi].
tan^2x=2secx-1

Answer 1

tan^2 x - 2 sec x +1

Answer 2

i simplified tan^2x into sec^2x-1. is that a good step?

Answer 3

(sec^2 x -1) - 2 sec x +1
sec^2 x - 2 sec x

Answer 4

sec (x)=u
so substituting
u^2-2u=0
We may complete the square by adding 1 to both side
u^2 -2u+1=1
(u-1)(u-1)=1
u=2
sec (x)=2 so since 1/sec(x)= cos(x)
cos(x)=1/2
x=pi/3

Answer 5

wait what is u?

Answer 6

I just define u for sec(x) for simplicity

Answer 7

ok so is the work with sec(x)=u a continuation of the post you did before that?

Answer 8

U is used in place of sec(x), just to make it easier to see

Answer 9

no, I did not need to use U. I could have instead solve it using sec(x)

Answer 10

Solved without using u

sec(x)^2-2sec(x)=0
We may complete the square by adding 1 to both side
sec(x)^2 -2sec(x)+1=1
(sec(x)-1)(sec(x)-1)=1
sec(x)=2
sec (x)=2 so since 1/sec(x)= cos(x)
cos(x)=1/2
x=pi/3

Answer 11

could i have used the quadratic formula to solve from where it was secx^2-2secx=0?

Answer 12

and also how do you know you have to complete the square?

Answer 13

Yes, but it is much easier

Answer 14

u^2-2u=0
I could see that by adding 1 to left side, I would be able to factor easily. But if you add something one one side, same must be done on the other side

Answer 15

thank you!