Find all solutions to tan^2x - secx = -1

Question

Find all solutions to the equation in the interval [0, 2pi]

Answer 1

First convert tan²x using trig identities, then simplify:

sin²x/cos²x - sec x = -1

(1-cos²x)sec²x - sec x + 1 = 0

sec²x - 1 - sec x + 1 = 0

sec²x - sec x = 0

sec x(sec x - 1) = 0

Then we have:

sec x = 0

sec x - 1 = 0

The first equation is invalid, so our only solutions come from the second equation. So, we have

sec x = 1

cos x = 1

x = 0, 2π

Answer 2

Recall that one of your pythagorean identities states that tan²(x) + 1 = sec²(x) Therefore, if you bring the -1 from the right side over to the left side, we get:

tan²(x) + 1 - sec(x) = 0

sec²(x) - sec(x) = 0

sec(x)(sec(x) - 1) = 0

sec(x) = 0 or sec(x) - 1 = 0

sec(x) = 0 or sec(x) = 1

But, sec(x) ≠ 0

Thus, sec(x) = 1 when

x = 0, 2π

Hope this helps

Mr. K