As #sec75=1/cos75", we have to first find the value of "cos75#.
We have to use the following Half-Angle Formula :
#cos(theta/2)=+-sqrt{(1+costheta)/2},#
where, the sign (+ or -) is to be determined in accordance with
#cos(theta/2)#
Taking, #theta/2=75, i.e., theta=150#, and, noting that #75# lies in the
First Quadrant , where, #cos" is "+ve#, we get,
#cos75=+sqrt((1+cos150)/2)=sqrt((1+cos(180-30)/2)#
#=sqrt((1-cos30)/2)=sqrt((1-sqrt3/2)/2)=sqrt((2-sqrt3)/4)#
#=1/2sqrt(2-sqrt3)=1/2sqrt(2-2sqrt(3/4))#
#=1/2sqrt(3/2+1/2-2sqrt(3/2*1/2))#
#=1/2sqrt{sqrt(3/2)^2+sqrt(1/2)^2-2*sqrt(3/2)*sqrt(1/2)}#
#=1/2sqrt{(sqrt(3/2)-sqrt(1/2))^2}#
#=1/2(sqrt(3/2)-sqrt(1/2))=(sqrt3-1)/(2sqrt2)#.
Therefore,
#sec75=1/cos75=(2sqrt2)/(sqrt3-1)#
#=(2sqrt2)/(sqrt3-1)xx(sqrt3+1)/(sqrt3+1)#
#=sqrt2(sqrt3-1)#
#=sqrt6-sqrt2#.
Enjoy Maths!
