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Question

Answers

A) $8\pi $

B) $4 \pi$

C) $\pi$

D) $2\pi$

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Given function is $f(x) = \sin (\cos x) + \cos (\sin x)$

We know that period of $\sin x,\cos x$ is $2\pi $

If we observe the terms in the given function they are in the form $f(g(x))$ which is a composite function.

We know that the period of a function of type $f(g(x))$ is the same as the period of g(x).

Now by using the above concept we can say that period of $\sin \left( {\cos x} \right)$ is $2\pi $ and the period of $\cos (\sin x)$ is also $2\pi $. Since period of $\sin x,\cos x$ is $2\pi $.

Here the given function f(x) is of the form $h(x) = p(x) + q(x)$.

We know that if any function is of the form $h(x) = p(x) + q(x)$ then the period of that function will be the L.C.M of periodic function of p(x) and q(x).

From the given function f(x) we can say that $p(x) = \sin (\cos )$ whose period is $2\pi $ and $q(x) = \cos (\sin x)$ whose period also $2\pi $.

Therefore by applying the above condition to the given function we can say that L.C.M of$2\pi ,2\pi = 2\pi $.

Hence the period of the function $f(x) = \sin (\cos x) + \cos (\sin x)$ is $2\pi $.

We can observe the period of individual functions in the below graphs.

Graph of $\sin x$

Graph of $\cos x$

Graph of $\sin(\cos x)$

Graph of $\cos (\sin x)$

Graph of $\sin (\cos x) + \cos (\sin x)$