the function y= x/tanx is ( an even, an odd, neither an even or odd) function, the function y= secx/x is (an even, an odd, neither an even or odd) function? I have to select one of the answers in parentheses.
Accepted Solution
A:
Answer:If [tex]v(x)=\frac{x}{\tan(x)}[/tex], then [tex]v[/tex] is even.If [tex]w(x)=\frac{\sec(x)}{x}[/tex], then [tex]w[/tex] is odd.Step-by-step explanation:Summary of rules/ what we need:[tex]f(-x)=f(x)[/tex] implies [tex]f[/tex] is even.[tex]f(-x)=-f(x)[/tex] implies [tex]f[/tex] is odd.So in either case, we need to replace [tex]x[/tex] with [tex]-x[/tex].Let's begin.First Problem:[tex]v(x)=\frac{x}{\tan(x)}[/tex]Replace [tex]x[/tex] with [tex]-x[/tex]:[tex]v(-x)=\frac{-x}{\tan(-x)}[/tex][tex]v(-x)=\frac{-x}{-\tan(x)}[/tex] (We used [tex]\tan(x)[/tex] is odd; that is, [tex]\tan(-x)=-\tan(x)[/tex])[tex]v(-x)=\frac{x}{\tan(x)}[/tex][tex]v(-x)=v(x)[/tex]This implies [tex]v[/tex] is an even function.Second Problem:[tex]w(x)=\frac{\sec(x)}{x}[/tex]Replace [tex]x[/tex] with [tex]-x[/tex]:[tex]w(-x)=\frac{\sec(-x)}{-x}[/tex][tex]w(-x)=\frac{\sec(x)}{-x}[/tex] (We used [tex]\sec(x)[/tex] is even; that is, [tex]\sec(-x)=\sec(x)[/tex])[tex]w(-x)=-\frac{\sec(x)}{x}[/tex][tex]w(-x)=-w(x)[/tex]This implies [tex]w[/tex] is an odd function.