Question

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

Answer 1

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)`

Answer 2

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)`