tan(x+45)=sin(x+45)/cos(x+45)=sin(x+45)sec(x+45).
sin(x+45)=sin(x)cos45+cos(x)sin45. sin45=cos45=√2/2.
sin(x+45)=(√2/2)(sin(x)+cos(x)), tan(x+45)=(√2/2)(sin(x)+cos(x))sec(x+45).
When x=0, tan(x+45)=tan(45)=1, (√2/2)(sin(x)+cos(x))sec(x+45)=(√2/2)(1)√2=1.
The given expression is squared 2×1×√2. Whatever squared 2 cos x means it has to be √2/2 or 1/√2, when x=0.
When x=45, tan(x+45)=tan90 which is undefined (infinite). sec90 is also undefined.
When x=-45, tan(x+45)=tan0=0, sec0=1, cos0=1. squared 2 cos x must equal 0 to make the supposed identity true. So squared 2 must be zero, which is not possible because it has to be a consistent constant. However, sin(-45)=-sin(45) and cos(-45)=cos(45), therefore sin(x)+cos(x)=0 when x=-45, so the identity as calculated holds.