The first thing to identify is how:
#1/(cos^2 x ) = sec^2 x #
Yielding:
#sec^2 x = 1- tanx#
Now we must use the identity:
#1 + tan^2 x = sec^2 x #
So hence:
#=> 1 + tan^2x = 1 - tanx #
#=> tan^2x + tanx = 0 #
#=> tanx ( tanx +1 ) = 0 #
# => tanx = 0 #, #tanx = -1 #
#--------------------#
( sub-lesson if you didnt already know: )
General solution for #tanx = tan beta #:
#x = pi n + beta #
or in degrees:
#tanx^circ= tan gamma^circ #
#x^circ = 180^circ n + gamma^circ #
#n in ZZ#
#--------------------#
Now to solve : #tanx = 0:#
#tanx = tan 0 #
#=> x = pin + 0 # Using the general solution to #tan x = tan beta #
Now to solve #tanx = -1 :#
#tanx = tan (-pi/4 ) #
#=> x = pi n -pi/4 # again using the general solution of #tanx #
In degrees:
# x = 180^circ n, x = 180^circn - 45^circ #
#n in ZZ#
Hope this helped!
