Question
Answers

The period of the function $f(x) = \sin (\cos x) + \cos (\sin x)$ equal to
A) $8\pi   $
B) $4 \pi$
C) $\pi$
D) $2\pi$

Answer
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Hint: To solve this problem we need to have knowledge about functions concept. We know that the period of any function which is in the form h(x) +g(x) is the L.C.M of the periodic function h(x) and g(x).

Complete step-by-step answer:
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 $.
So, Option (D) is the correct answer.

Note: In this type of problems we won’t observe the given function clearly and directly apply the condition to get the period of the function which is a very wrong process. Here in the given question the function f(x) has one more function in the term of function. So here before finding the period of the function f(x) we have to find the period of the innermost function. In this case the innermost function is a composite function.
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 $\sin (\cos x) + \cos (\sin x)$