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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As far as I know, its due to how often they appear across maths – it helps to define them as their own thing(s) due to how frequently you have to write or use them. For example, they appear in the trig identities tan^2 x + 1 = cosec^2 x & 1 + cot^2 x = sec^2 x, and also often in calculus, like for example when considering d/dx(tanx) = sec^2 x. You could make the argument that anything in maths can be written as its underlying mathematical representation rather than being a separate entity with its own name, but as soon as it becomes frequently used, it simply becomes a matter of convenience to name it. Sec, csc, and cotan are no different.
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