Question

Even-Odd (a) Show that cot \(x\) is an odd function of \(x\) . (b) Show that the quotient of an even function and an odd function is an odd function.

Short answer

(a) The cotangent function is proved to be an odd function because it satisfies the definition of the odd function, cot(-x) = -cot(x). (b) The quotient of an even function and an odd function is an odd function, as it satisfies h(-x) = -h(x).

Step-by-step solution

Determine the nature of cot x

An odd function is defined as a function that fulfills the property f(-x) = -f(x). The cotangent of x can be expressed as cos x / sin x. So the step is to evaluate cot(-x), which gives cos(-x) / sin(-x) = cos x / -sin x = - cot x.

Prove cot x oddness

As we have shown that cot(-x) = -cot x, it can be concluded that the cotangent function is an odd function of x because it satisfies the definition of odd functions.

Determine the nature of the quotient of an even and odd function

An even function is defined as a function that fulfills the property f(-x) = f(x). So let's take one function that is even, f(x), and another that is odd, g(x). We want to prove that the quotient h(x) = f(x) / g(x) is an odd function.

Prove the quotient to be an odd function

To achieve this, we evaluate the function h(-x) = f(-x) / g(-x). By properties of even and odd functions: h(-x) = f(x) / -g(x) = -f(x) / g(x) = -h(x). This proves that the quotient of an even and odd function is an odd function.