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

1 Answer
Nghi N.
Sep 19, 2015

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

Explanation:

Call #tan (x/2) = t#. Use the formulas: #cos x = (1 - t^2)/(1 + t^2)# and
#tan x = (2t)/(1 - t^2).#
cos x + tan x = 1
#(1 - t^2)/(1 + t^2) + (2t)/(1 - t^2) = 1#
#(1 - t^2)^2 + 2t(1 + t^2) = 1 - t^4#
#(1 + t^4 - 2t^2) + 2t + 2t^3 = 1 - t^4#
#t^4 + t^3 - t^2 + t = 0#
#t(t^3 + t^2 - t + 1) = 0#
#t = tan (x/2) = 0# --> #x/2 = 0# and #x/2 = pi# --> #x = 0#
and #x = 2pi#
(t^3 + t^2 - t + 1) = 0
Solve by graphing calculator.

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