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

1 Answer
Nghi N.
Sep 26, 2016

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

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