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!
