Question

Find the period of $\cos (\cos x)+\cos (\sin x)?$

Answer 1

Hint: The length of one complete cycle is called a period of a trigonometric function. The cosine function is even and the sine function is odd. A function is even, if $g(-x)=g\left(x\right).$ A function is odd, if $g(-x)=-g\left(x\right).$

Complete step-by-step solution:
We know that the period of a trigonometric function is the length of one complete cycle.
Consider the given trigonometric function $\cos (\cos x)+\cos (\sin x).$
Now, let us suppose that $f\left(x\right)=\cos (\cos x)+\cos (\sin x).$
Also, we know that the cosine function is even and the sine function is odd.
We have already learnt that, for an even function $g\left(x\right),g(-x)=g\left(x\right).$
Similarly, for an odd function $h\left(x\right),h(-x)=-h\left(x\right).$
So, we will get $\cos (-x)=\cos x$ and $\sin (-x)=-\sin x.$
Now we use these identities to find the period of the given function $\Rightarrow f\left(x\right)=\cos (\cos x)+\cos (\sin x).$
 Also, we know that $\cos ({\displaystyle \frac{\pi }{2}}+x)=-\sin x.$
Similarly, $\sin ({\displaystyle \frac{\pi }{2}}+x)=\cos x.$
Because, in the second quadrant, the sine function is positive and the cosine function is negative.
Since we have these identities, we can find that
$\Rightarrow f({\displaystyle \frac{\pi }{2}}+x)=\cos (\cos ({\displaystyle \frac{\pi }{2}}+x))+\cos (\sin ({\displaystyle \frac{\pi }{2}}+x)).$
From the above, we will get the following
$\Rightarrow f({\displaystyle \frac{\pi }{2}}+x)=\cos (-\sin x)+\cos (\cos x).$
Now we can use the properties of even functions to get,
$\Rightarrow f({\displaystyle \frac{\pi }{2}}+x)=\cos (\sin x)+\cos (\cos x).$
Because, we will get $\cos (-\sin x)=\cos (\sin x)$ and $\cos (\cos x)=\cos (\cos x).$
And now we can see that
$\Rightarrow f({\displaystyle \frac{\pi }{2}}+x)=\cos (\cos x)+\cos (\sin x).$
And this will give us the following
$\Rightarrow f({\displaystyle \frac{\pi }{2}}+x)=\cos (\cos x)+\cos (\sin x)=f\left(x\right).$
Thus, we will get $f({\displaystyle \frac{\pi }{2}}+x)=f\left(x\right).$
Hence, we can conclude that the period of the given function $f\left(x\right)=\cos (\cos x)+\cos (\sin x)$ is ${\displaystyle \frac{\pi }{2}}.$

Note: The period of the cosine function is $2\pi .$ That is, the period of $\cos x=2\pi .$ The period of the sine function is also $2\pi .$ That is, the period of $\sin x=2\pi .$ The period can be found as follows:
Period of $\cos (\cos x)=\pi$ and $\cos (\sin x)=\pi .$
Now the period of $f\left(x\right)={\displaystyle \frac{1}{2}}\left(\text{LCM of}\pi \text{ and }\pi \right)$
That is, $f\left(x\right)={\displaystyle \frac{\pi }{2}},$ Since $\text{LCM of }\pi \text{ and }\pi =\pi .$