Why does this not work??? (Trig)

Sin^2x + Cos^2x = 1

that is a good question why don't we square sinx+cosx and see what happens.  

$$\\(Sinx + Cosx)^2 \\\\
=sin^2x+cos^2x+2sinxcosx\\\\
=1+2sinxcosx$$

So

$$\\(sinx+cosx)^2=1+2sinxcosx\\\\
$Taking the square root of both sides we get $\\\\
sinx+cosx=\pm\sqrt{1+2sinxcosx}$$

It can only equal one if   2sinxcosx=0       and that doesn't happen very often.        

√(x²+y²) ≠ x + y       (Except for rare cases.)

For instance: √(3²+4²) = √(9+16) = √25 = 5

But: if √(x²+y²) = x + y, then √(3²+4²) would equal 3 + 4 = 7 (which it doesn't).