How do you prove # tan^2x-1 = 1+tanx #?
2 Answers
The given equation is not true!
Explanation:
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
Explanation:
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.
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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#
Substitute
#y^2 - y - 2 = 0#
... solve the quadratic equation...
#y = 2 " or " y = -1# .
Substitute back:
#tan x = 2 " or " tan x = -1#
Thus, the solutions are:
#x = arctan (2) " or " x = arctan(-1)#