Question 241174
Whenever a Trig problem refers to "exact values" you will end up using angles that are multiples of 30 or 45 degrees (or {{{pi/6}}} or {{{pi/4}}} radians).<br>
Since you are looking for the exact value of {{{sec(5pi/12)}}} we have to find a way to express {{{5pi/12}}} in terms of multiples of {{{pi/6}}} and/or {{{pi/4}}}. This expression can be a sum or a difference. The expression can be twice or half a special angle. And this expression can be some combination of any of these.<br>
So what expression of multiples of {{{pi/6}}} and/or {{{pi/4}}} equals {{{5pi/12}}}? There are probably others but the one that is most obvious to me is: {{{(1/2)(5pi/6) = 5pi/12}}}. SO we will be using half-angle formulas.<br>
There is no half-angle formula for sec, so we will find {{{cos(5pi/12)}}} and then find its reciprocal to get the sec:
{{{cos((1/2)(5pi/6)) = sqrt((1 + cos(5pi/6))/2)}}}
(The {{{cos((1/2)x)}}} formula has a "+-" in front of the square root. However, since {{{5pi/12 < pi/2}}}, {{{5pi/12}}} is in the first quadrant.  And since {{{5pi/12}}} is in the first quadrant where all Trig functions are positive, we will discard the "-" and use the positive square root.)<br>
Now we just find {{{cos(5pi/6)}}}, substitute and simplify. {{{5pi/6}}} is in the second quadrant where cos is negative. And its reference angle is {{{pi/6}}}. Together this makes {{{cos(5pi/6) = (-sqrt(3)/2)}}}. Substituting this into our equation we get:
{{{cos((1/2)(5pi/6)) = sqrt((1 + (-sqrt(3)/2))/2)}}}
To simplify this we will start by eliminating the fraction within a fraction. We can do this my multiplying the numerator and denominator of the "big" fraction by the LCD of the "little" fraction(s). Since there is only one "little" fraction, we will use its denominator, 2:
{{{cos((1/2)(5pi/6)) = sqrt(((1 + (-sqrt(3)/2))/2)(2/2))}}}
which simplifies to:
{{{cos((1/2)(5pi/6)) = sqrt((2 + (-sqrt(3)))/4)}}}
Using the property, {{{sqrt(a/b) =  sqrt(a)/sqrt(b)}}}:
{{{cos((1/2)(5pi/6)) = sqrt((2 + (-sqrt(3))))/sqrt(4)}}}
which simplifies to:
{{{cos(5pi/12) = sqrt(2 + (-sqrt(3)))/2}}}
Now we can find the sec by finding the reciprocal of cos:
{{{sec(5pi/12) = 2/sqrt(2 + (-sqrt(3)))}}}
This may be an acceptable answer. However, the denominator is not rational. If you want rationalized denominators:
{{{sec(5pi/12) = (2/sqrt(2 + (-sqrt(3))))((sqrt(2 + (-sqrt(3))))/(sqrt(2 + (-sqrt(3)))))}}}
{{{sec(5pi/12) = 2*sqrt(2 + (-sqrt(3)))/(2 + (-sqrt(3)))}}}
{{{sec(5pi/12) = (2*sqrt(2 + (-sqrt(3)))/(2 + (-sqrt(3))))((2+sqrt(3))/(2+sqrt(3)))}}}
{{{sec(5pi/12) = (2*sqrt(2 + (-sqrt(3)))*(2+sqrt(3)))/(2^2 - (sqrt(3))^2)}}}
{{{sec(5pi/12) = (2*sqrt(2 + (-sqrt(3)))*(2+sqrt(3)))/(4 - 3)}}}
{{{sec(5pi/12) = (2*sqrt(2 + (-sqrt(3)))*(2+sqrt(3)))/1}}}
{{{sec(5pi/12) = 2*sqrt(2 + (-sqrt(3)))*(2+sqrt(3))}}}
I'll leave it up to you to multiply this out.