How do you prove # tan^2x-1 = 1+tanx #?
The given equation is not true!
As a counter example, consider
It's not an identity, so it can't be proven.
However, the equation can be solved and has the solutions
You can't prove this because it isn't an identity.
If it was an identity, you would have:
#tan^2 x - 1 = (tan x + 1)(tan x -1) stackrel("? ")(=) 1 + tan x#
which could only be true if
This is certainly not the case.
However, even though you can't prove this as an identity (valid for all
Let's do this.
#tan^2 x - 1 = 1 + tan x #
#<=> tan^2 x - tan x - 2 = 0#
#y^2 - y - 2 = 0#
... solve the quadratic equation...
#y = 2 " or " y = -1#.
#tan x = 2 " or " tan x = -1#
Thus, the solutions are:
#x = arctan (2) " or " x = arctan(-1)#