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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It’s just a convention. You can work without them, but they have historical reasons for existing and some people are going to use them whether you like them or not, so you need to know them. There are also some situations when it’s much easier to use them than to fiddle around with fraction notation.
Also, if you want the reciprocal of sin(x), writing cosec(x) is much safer than using exponents to write (sin(x))^(-1), because that has the risk of being mixed up with sin^(-1)(x), which is the *inverse sin* of x, not the reciprocal of the sin of x.
“But we use sin^(2)(x) to mean the square of sin(x), not the sin of the sin of x, so why is sin^(-1)(x) different?” you might ask, and you might think that that’s terrible ambiguous notation, and you’d be right. Sometimes bad notation gets entrenched in the way people do maths, and there’s nothing you can do about it except strive for clarity yourself, and cosec(x) helps you avoid ambiguity.
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