How do you prove # tan^2x-1 = 1+tanx #?

2 Answers
Feb 10, 2016

The given equation is not true!

Explanation:

As a counter example, consider #x=0#
#color(white)("XXX")tan(x=0) = 0#
#color(white)("XXX")tan^2(x=0) = 0#

#tan^2(x=0)-1=-1 != +1 =1+tan(x=0)#

Feb 10, 2016

It's not an identity, so it can't be proven.

However, the equation can be solved and has the solutions #x = arctan(2)# or #x = arctan(-1)#.

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 #tan x - 1 = 1# was true for every #x#.

This is certainly not the case.

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However, even though you can't prove this as an identity (valid for all #x#), you can solve the equation and find some #x# as solutions.

Let's do this.

#tan^2 x - 1 = 1 + tan x #

#<=> tan^2 x - tan x - 2 = 0#

Substitute #y = tan x#:

#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)#