Question

How do you solve `sec^2x+tanx-1=0`?

Answer 1

`0, (3pi)/4, pi, (7pi)/4, 2pi`

Explanation:

There are 2 variables: sin x and cos x. General Method: we must transform the trig equation into a product of 2 simple trig equations.
`1/(cos^2 x) + sin x/(cos x) = 1`
`1 + sin x.cos x = cos^2 x`
`(1 - cos^2 x) + sin x.cos x` = 0
`sin^2 x + sin x.cos x = 0`
`sin x(sin x + cos x) = 0`
Now, we solve the two simple trig equations.
a. sin x = 0 --> `x = 0, and x = pi, and x = 2pi`
b. Use trig identity:
`sin a + cos a = sqrt2cos (a - pi/4)`
`sin x + cos x = sqrt2cos (x - pi/4) = 0`
`cos (x - pi/4) = 0`
Trig unit circle -->
c. `x - pi/4 = pi/2` -->
`x = pi/2 + pi/4 = (3pi)/4`
d. `x - pi/4 = (3pi)/2` -->
`x = (3pi)/2 + pi/4 = (7pi)/4`
Answers for `(0, 2pi)`
`0, (3pi)/4, pi, (7pi)/4, 2pi`