This is one of those rare questions that you can evaluate exactly using the sum and différence formulas.
First, though, let's define #sectheta#. By the reciprocal identities #sectheta = 1/costheta#
#sec15#
#=1/cos15#
Now, #15^@# can be written as #60^@ - 45^@#
By the sum and différence identity #cos(alpha - theta) = cosalphacostheta + sinalphasintheta#
We can therefore state the following:
#1/cos15 = 1/cos(60 - 45)#
Expanding:
#=1/(cos60cos45 + sin60sin45)#
#=1/(1/2 xx 1/sqrt(2) + sqrt(3)/2 xx 1/sqrt(2))#
#= 1/((1/(2sqrt(2)) + sqrt(3)/(2sqrt(2)))#
#= 1/((1 + sqrt(3))/(2sqrt2))#
#= (2sqrt(2))/(1 + sqrt(3))#
Rationalizing the denominator:
#= (2sqrt(2))/(1 + sqrt(3)) xx (1 - sqrt(3))/(1 - sqrt(3))#
#=(2sqrt(2) - 2sqrt(6))/-2#
#=(2(sqrt(2) - sqrt(6)))/-2#
#= sqrt6 - sqrt(2)#
Therefore, #sec15 = sqrt(6) - sqrt(2)#
Hopefully this helps!