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) = 11cos2x+sinxcosx=1
1 + sin x.cos x = cos^2 x1+sinx.cosx=cos2x
(1 - cos^2 x) + sin x.cos x(1−cos2x)+sinx.cosx = 0
sin^2 x + sin x.cos x = 0sin2x+sinx.cosx=0
sin x(sin x + cos x) = 0sinx(sinx+cosx)=0
Now, we solve the two simple trig equations.
a. sin x = 0 --> x = 0, and x = pi, and x = 2pix=0,andx=π,andx=2π
b. Use trig identity:
sin a + cos a = sqrt2cos (a - pi/4)sina+cosa=√2cos(a−π4)
sin x + cos x = sqrt2cos (x - pi/4) = 0sinx+cosx=√2cos(x−π4)=0
cos (x - pi/4) = 0cos(x−π4)=0
Trig unit circle -->
c. x - pi/4 = pi/2x−π4=π2 -->
x = pi/2 + pi/4 = (3pi)/4x=π2+π4=3π4
d. x - pi/4 = (3pi)/2x−π4=3π2 -->
x = (3pi)/2 + pi/4 = (7pi)/4x=3π2+π4=7π4
Answers for (0, 2pi)(0,2π)
0, (3pi)/4, pi, (7pi)/4, 2pi0,3π4,π,7π4,2π
