JEE Main 2002 — Functions Question with Solution

$$\sqrt{x}$$ is non periodic function and $$\cos \left(something\right)$$ is a periodic function so here in $$\cos \sqrt{x}$$ $$\to$$ inside periodic function there is non periodic function which always produce non periodic function.

$${\cos }^{2}x$$ is a periodic function with period $$\pi$$

Note :
(1) When $$n$$ is odd then the period of $${\sin }^n\theta$$, $${\cos }^n\theta$$, $${\csc }^n\theta$$, $${\sec }^n\theta$$ = $$2\pi$$

(2) When $$n$$ is even then the period of $${\sin }^n\theta$$, $${\cos }^n\theta$$, $${\csc }^n\theta$$, $${\sec }^n\theta$$ = $$\pi$$

(3) When $$n$$ is even/odd then the period of $${\tan }^n\theta$$, $${\cot }^n\theta$$ = $$\pi$$

(3) When $$n$$ is even/odd then the period of $$\left|{\sin }^n\theta \right|$$, $$\left|{\cos }^n\theta \right|$$, $$\left|{\csc }^n\theta \right|$$, $$\left|{\sec }^n\theta \right|$$, $$\left|{\tan }^n\theta \right|$$, $$\left|{\cot }^n\theta \right|$$ = $$\pi$$

$$\cos \sqrt{x}+{\cos }^{2}x$$ = non periodic function + periodic function = non periodic function