90-67.5=22.5=45/2.
TAN45=1 and TAN45=2TAN22.5/(1-TAN²22.5)=1
Let t=TAN22.5: 2t=1-t², t²+2t=1, t²+2t+1=2, (t+1)²=2, t=-1±√2. TAN22.5=√2-1.
TAN67.5=COT22.5=1/(√2-1)=√2+1.
Consider a right triangle with sides √2+1, 1 and √((√2+1)²+1)=√(4+2√2).
SIN67.5=(√2+1)/√(4+2√2); COS67.5=1/√(4+2√2).