The function is .

Use the mean value theorem to show that the function has atleast one real root.

Mean value Theorem :

If is continuous on  and differentiable on open interval , then there exists a number  in .

The function is continuous and differentiable.

.

.

So the function have atleast one real root on open interval .

Use the Rolles theorem to show that the function has only one real root.

Rolles Theorem :

Let  be a function that satisfies the following three hypotheses.

1.  is continuous on .

2.  is differentiable on .

3. .

Then there is a number  in  such that .

The function is continuous and differentiable.

Suppose , are distinct real numbers such that .

Apply derivative with respect to .

From the Rolles theorem number  in  such that .

Range of the cosine function is to .

Therefore the function does not have roots while equting .

But according to Mean value theorem the remaining one root is real.

It is clear that the functon have only real root.

The function has exactly one real solution.