Question 1205933
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Short answer:  A function can only return one output value for each input value.<br>
Since trigonometric functions are periodic, the domain for each inverse function must be restricted to an interval on which each trigonometric function takes on all of its values exactly once.<br>
The sine and cosine functions (and their reciprocal functions cosecant and secant)   take on all their values over one-half of their period, so their inverse functions have to be restricted to an interval equal to half the period of the sine and cosine functions.<br>
The tangent and cotangent functions take on all their values once over their whole period, so their inverse functions need to be restricted to a period equal to the period of the function.<br>