How do you evaluate ##sec(cos^-1(1/2))## without a calculator?

##sec(cos^(-1)(1/2)) = 2##

Another way, without calculating ##cos^(-1)(1/2)##:

Consider a right triangle with an angle ##theta = cos^(-1)(1/2)##. Then ##cos(theta) = 1/2##, meaning the ratio of its adjacent side to the hypotenuse is ##1/2##. Thus the ratio of the hypotenuse to its adjacent side, that is, ##sec(theta)##, is ##2/1 = 2##.

Thus ##sec(cos^(-1)(1/2)) = sec(theta) = 2##

Note that this same reasoning shows that in general, ##sec(cos^(-1)(x)) = 1/x##

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