How do you find the exact functional value sec 1275 using the cosine sum or difference identity?

Find the value of sec (1275)

Ans:## - (2sqrt2)/(1 + sqrt3)##

##sec (1275) = 1/(cos 1275)##. First, find ##cos (1275).## ##cos 1275 = cos (195 + 1080) = cos (195 + 3(360)) = cos (195).## ##cos 195 = cos (60 + 135) = cos 60.cos 135 - sin 60.sin 135 =## ##= (1/2)(-sqrt2/2) - (sqrt3/2)(sqrt2/2) = - (sqrt2/4)(1 + sqrt3)##

##sec 1275 = 1/(cos 195) = - 4/((sqrt2)(1 + sqrt3)) = -(2sqrt2)/(1 + sqrt3##

Check by calculator. cos 195 = -0.97 cos 195 = - (1 + sqrt3)/(2sqrt2) = -(2.73)/(2.82) = - 0.97. OK