sec 292.5 = sec (585/2) = 1/cos(585/2)sec292.5=sec(5852)=1cos(5852)
Let us calculate cos (585/2)cos(5852)
cos (585/2) = cos [(720-135)/2] = cos(360 - 135/2)cos(5852)=cos[720−1352]=cos(360−1352)
= cos(-135/2) = cos(135/2)=cos(−1352)=cos(1352). ----------(1.)
Now,
cos (x/2) = +-sqrt[(1+cosx)/2]cos(x2)=±√1+cosx2
therefore cos(135/2) = +-sqrt[(1+cos135)/2]
cos 135 = cos (180 - 45) = -cos 45 = -1/sqrt2 = -0.71
=> cos(135/2) = +-sqrt[(1-0.71)/2]
= +- sqrt(0.29/2) = +-sqrt(.0145) = +-0.381
but 135/2 = 67.5 < 90. Hence cos(135/2) will be positive.
therefore cos(135/2) = 0.381
From (1.) cos (585/2) = cos(135/2) = 0.381
=> sec 292.5 = sec (585/2) = 1/cos(585/2) = 1/.381 = color(red)2.625
If you need help with the half angle formula, do comment below.