Trigonometry/Algebra

First, you've got to know that:
sin(a) = O/H
cos(a) = A/H
tan(a) = O/A
cot(a) = A/O

Therefore:
sin(a)/cos(a) = O/A = tan(a)
cos(a)/sin(a) = A/O = cot(a)
tan(a) = 1/cot(a)
cot(a) = 1/tan(a)

With:
A: Adjacent side
O: Opposite side
H: Hypotenuse

cos(x) + sin(x) =
cos(x)/(1-tan(x)) + sin(x)/(1-cot(x)) =
cos(x)/[1-sin(x)/cos(x)] + sin(x)/[1-cos(x)/sin(x)] =
cos(x)/[cos(x)/cos(x) - sin(x)/cos(x)] + sin(x)/[sin(x)/sin(x) - cos(x)/sin(x)] =
cos(x)/[(cos(x)-sin(x))/cos(x)] + sin(x)/[(sin(x)-cos(x))/sin(x)]
Hence: 1 = sin(x)/sin(x) = cos(x)/cos(x)

cos(x)/[(cos(x)-sin(x))/cos(x)] + sin(x)/[(sin(x)-cos(x))/sin(x)] =
cos(x)*cos(x)/[cos(x)-sin(x)] + sin(x)*sin(x)/[sin(x)-cos(x)] =
cos(x)²/[cos(x)-sin(x)] + (-1)*sin(x)²/[cos(x)-sin(x)] =
cos(x)²/[cos(x)-sin(x)] - sin(x)²/[cos(x)-sin(x)] = [cos(x)²-sin(x)²]/[cos(x)-sin(x)] =
cos(x)+sin(x) (* [cos(x)-sin(x)])

=> cos(x)²-sin(x)² = [sin(x)+cos(x)]*[cos(x)-sin(x)] = cos(x)²-sin(x)²-sin(x)*cos(x)+sin(x)*cos(x) = cos(x)²-sin(x)²

Vielen Dank!