You are assuming incorrectly. Those are NOT the buttons you need to press. It is likely that your calculator does not have a one-step button for such rarely-used functions as CSC, SEC, and COT. However, "Sin-1" is "the angle whose sine is." It is entirely different from the inverse trig function CSC.
Remember, sin = opp/hyp. CSC= hyp/opp. Draw out a right triangle and go back to the basics to be sure you understand them!!
To get CSC, you take the sine of the argument, then take the result and 1/x it !! CSC=1/sin.
Totally different interpretation than sin-1. Your calculator's "sin-1" DOES NOT mean ""sine" to the minus-one power," it means "the angle whose sine is...!!"
Same with cos. Cos = adj/hyp. SEC= hyp/adj. SEC= 1/cos.
To get SEC, take the cosine, then take the result, hit the "one over x" key (I'm assuming the calculator has one), and you have that!
The angle whose sine is 60 is in fact meaningless, since sine is always between zero and one inclusive. Your calculator is functioning as it ought to, when it gives you an error message for putting 60 in as an argument for "sin-1." There is no such angle whose sine is 60.
Cotangent: take the tangent, and hit 1 / X . COT= adj/opp=1/tan.
Hope that helped!