Question

Question is in the pictures.

Answer 1

x/tan(x) is an even function

sec(x)/x is an odd function

Step-by-step explanation:

x/tan(x)

For f(x) = x/tan(x), consider f(-x).

... f(-x) = -x/tan(-x)

Now, we know that tan(x) is an odd function, so tan(-x) = -tan(x). Using this, we have ...

... f(-x) = -x/(-tan(x)) = x/tan(x) = f(x)

The relation f(-x) = f(x) is characteristic of an even function, one that is symmetrical about the y-axis.

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sec(x)/x

For g(x) = sec(x)/x, consider g(-x).

... g(-x) = sec(-x)/(-x)

Now, we know that sec(x) is an even function, so sec(-x) = sec(x). Using this, we have ...

... g(-x) = sec(x)/(-x) = -sec(x)/x = -g(x)

The relation g(-x) = -g(x) is characeristic of an odd function, one that is symmetrical about the origin.