Yes, other Trig functions and their derivatives exist.
The first six are:
d/dx sin(x) = cos(x)
d/dx cos(x) = - sin(x)
d/dx tan(x) = sec2(x)
d/dx sec(x) = sec(x)tan(x)
d/dx csc(x) = - csc(x)cot(x)
d/dx cot(x) = - csc2(x)
I like to group them as follows, with those without "co" in the name in the 1st column and those with "co" in the 2nd. We can see patterns to help memorize them better this way.
d/dx sin(x) = cos(x) d/dx cos(x) = - sin(x)
d/dx sec(x) = sec(x)tan(x) d/dx csc(x) = - csc(x)cot(x)
d/dx tan(x) = sec2(x) d/dx cot(x) = - csc2(x)
(This is intended to be in two columns, but the Wyzant software keeps taking the space out)
The trig functions are based off of a unit circle, with equation x2 + y2 = 1
Additionally there are 6 hyperbolic functions, which are based off of the hyperbola x2 - y2 = 1
Their derivatives are:
d/dx sinh(x) = cosh(x)
d/dx cos(x) = sinh(x)
d/dx tan(x) = sech2(x)
d/dx sech(x) = - sech(x)tanh(x)
d/dx csch(x) = - csch(x)coth(x)
d/dx coth(x) = - csch2(x)
Finally, all 12 of the functions above have inverse functions, and each of those have derivatives as well. They are much more complicated to write out, as most have a numerator, denominator, and root sign, so I would suggest a quick web search for "inverse trig derivatives" and inverse hyperbolic derivatives".
Hope that helps.