sec(1275°).
1275/360=3+13/24.
13/24×360=195°. Therefore 1275=3×360+195.
195=180+15, cos(30)=√3/2=2cos2(15)-1.
cos(180+15)=-cos(15)=-1/sec(15), sec(1275)=sec(195)=-sec(15).
cos2(15)=½(1+√3/2)=(2+√3)/4,
sec2(15)=4/(2+√3)=4(2-√3)=8-4√3,
sec(15)=√(8-4√3).
Let √a-√b=√(8-4√3), then, squaring, a+b-2√(ab)=8-4√3,
a+b=8, 2√(ab)=4√3, squaring: 4ab=48, ab=12, b=12/a, a+12/a=8, a2-8a+12=0=(a-6)(a-2).
If a=6, 6b=12, so b=2 and sec(1275)=sec(195)=-sec(15)=-(√6-√2).
Therefore, sec(1275)=√2-√6=-1.0353 approx.