(i) (\sin^2 x)

( f(-x) = \sin^2(-x) = (-\sin x)^2 = \sin^2 x = f(x) ) → even.

(ii) (\sin x + \cos x)

( f(-x) = -\sin x + \cos x \neq \pm f(x) ) → neither.

(iii) (\sin^4 x + \cos^4 x)

Even (sum of even powers).

(iv) (\tan x + \sec x)

Odd + even → neither.

(v) (\dfrac{1}{\operatorname{cosec}^3 x} = \sin^3 x) → odd.

(vi) (\dfrac{\sin x + \sin 3x}{\cos x + \cos 3x}) → simplifies to (\tan 2x) (odd).

(vii)–(viii) Check parity similarly.

Answers: even; neither; even; neither; odd; odd; even; neither (approximate).