Note that cos 75 = cos (45+30). We make this substitution because we know (or should know) the trig functions of 45 and 30 degrees).
cos (a+b) = cos a cos b - sin a sin b, so, therefore,
cos (45+30) = cos 45 cos 30 - sin 45 sin 30
= 1/sqrt(2) * (sqrt(3)/2) - 1/sqrt(2) * (1/2)
As you can see, we have sqrt(2) in the denominators of both terms.
sqrt(3) 1 sqrt(3) - 1
We get: cost 75 = ------------- - ------------- = ---------------
2sqrt(2) 2sqrt(2) 2sqrt(2)
"Rationalizing the denominator" here means to get rid of the radical sqrt(2) from the denominator of this final result. We want only integers in the denominator.
To achieve this, multiply both the numerator and the denom. of
sqrt(3) - 1
--------------- by sqrt(2). The result will be:
2sqrt(2)
sqrt(2)*[sqrt(3) - 1]
------------------------- , which has the real, integer denominator 4.
2(2)
You could leave this answer as is, or you could write it as
sqrt(6) - sqrt(2)
---------------------
4
