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.
Answer:
y = x/tanx is neither odd nor even
y=secx/x is an odd function
Step-by-step explanation:
these are the answers for PLATO
