Given sin x + cos x = 1, what is tan(x+x)?

We'll start by square raising the constraint from
enunciation:

 sinx + cosx =
1

(sinx + cosx)^2 =
1^2

(sinx)^2 + (cosx)^2 + 2sinx*cosx = 1
(1)

But, from the fundamental formula of
trigonometry:

(sinx)^2 + (cosx)^2 =
1

We'll substitute (sinx)^2 + (cosx)^2 by
1:

The relation (1) will
become:

1 + 2sinx*cosx =
1

We'll eliminate like
terms:

 2sinx*cosx =
0

But  2sinx*cosx = sin
(2x)

We'll write the formula for tan(x+x) =
tan 2x:

tan 2x = sin 2x/cos
2x

Since sin 2x = 0, we'll
get:

tan 2x = 0/cos
2x

tan (x+x) = tan 2x =
0