Trig problem

Q:  Find all solutions of the equation in the interval [0, 2pi )

tan²x – sec x = -1

(MULTIPLE CHOICE)

a. 0
b. pi/4 , 3pi/4 , 5pi/4 . 7pi/4
c. pi/6 , 5pi/6
d. 2pi/3 , pi , 4pi/3

A:

Let’s start with the equation:

tan²x – sec x = -1

Now, I see we have tan’s and sec’s mixed.  We don’t like this… We want to have only one trig funciton (makes life easier)…. I do recall a trig identity we can use:

tan²x + 1 = sec²x

Manipulate this to get:

tan²x = sec²x – 1

Substitute this in to our original equation:

tan²x – sec x = -1

(sec²x – 1) – sec x = -1

sec²x – 1 – sec x = -1

Add 1 to both sides:

sec²x – sec x = 0

Factor:

sec x (sec x – 1) = 0

So, either sec x = 0 or sec x – 1 = 0

(1)  sec x = 0 never… that never happens… so, we can through that out…

(2)  sec x – 1 = 0

sec x = 1

1 / cos x = 1

cos x = 1

Where does cos x = 1??

Refer to the exact values chart to find that!

cos x = 1 when x = 0.

So, the correct answer is a