# 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)#