Parity of a function

Calculus Level 3

What is the parity of $f(x) = \cos^2 x + \csc^3 x + \sin x^2?$

Even Neither even nor odd Odd

1 solution

Munem Shahriar
Jan 27, 2018

Condition for being even: $f(-x) = f(x)~ \forall x \in \mathbb{R}$

Condition for being odd: $f(-x) = -f(x) ~ \forall x \in \mathbb{R}$

Case 1:

$f(-x) = \cos^2 (-x) + \csc^3 (-x) + \sin x^2$

Case 2:

$-f(x) = -\cos^2 x - \csc^3 x - \sin x^2$

Since $f(-x) \ne f(x)$ and $-f(x) \ne f(-x),$ the function is neither even nor odd.