What is the purpose, if any, of secant, cosecant, and cotangent? I’m an engineering student so I use sine, cosine, and tangent a lot, but almost never the other 3. Creating a new trig function just to be the reciprocal seems to be a bit redundant.
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The most basic place where sec(x) occurs in math if you at first only care about sin(x), cos(x), and tan(x) is in the derivative formula
(tan x)’ = sec^(2)(x).
Trigonometric functions have hyperbolic trig analogues (essentially replace x with ix) and the hyperbolic trig function
sech(pi x) = 2/(e^(pi x) + e^(-pi x)) is interesting in Fourier analysis since it is equal to its Fourier transform (using a suitable normalization of that transform).
In higher math, I can think of two places where the function cot(x) has a purpose:
1. The Riemann zeta function can be calculated at positive even numbers by computing a series expansion for cot(x) in two ways.
2. The q-expansion of Eisenstein series for the group SL(2,Z) is derived using the function pi cot(pi x).
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