How do you solve - cos(x) -tan(x)= -1cos(x)tan(x)=1?

1 Answer
Sep 19, 2015

Solve f(x) = - cos x - tan x = - 1

Explanation:

Call tan (x/2) = ttan(x2)=t. Use the formulas: cos x = (1 - t^2)/(1 + t^2)cosx=1t21+t2 and
tan x = (2t)/(1 - t^2).tanx=2t1t2.
cos x + tan x = 1
(1 - t^2)/(1 + t^2) + (2t)/(1 - t^2) = 11t21+t2+2t1t2=1
(1 - t^2)^2 + 2t(1 + t^2) = 1 - t^4(1t2)2+2t(1+t2)=1t4
(1 + t^4 - 2t^2) + 2t + 2t^3 = 1 - t^4(1+t42t2)+2t+2t3=1t4
t^4 + t^3 - t^2 + t = 0t4+t3t2+t=0
t(t^3 + t^2 - t + 1) = 0t(t3+t2t+1)=0
t = tan (x/2) = 0t=tan(x2)=0 --> x/2 = 0x2=0 and x/2 = pix2=π --> x = 0x=0
and x = 2pix=2π
(t^3 + t^2 - t + 1) = 0
Solve by graphing calculator.