Question

F(x) =ln(cosx) is Even, Odd or periodic nature

• A. Even
• B. Odd
• C. Periodic

Answer 1

Answer: (A), (C)

Even, Periodic

Answer 2

To determine the nature of the function $F(x) = \ln(\cos x)$, we need to check its domain and then evaluate if it's even, odd, or periodic.

1. Domain of the function:

For $F(x) = \ln(\cos x)$ to be defined, two conditions must be met:

• $\cos x$ must be defined, which is true for all real $x$.
• The argument of the logarithm must be positive: $\cos x > 0$.

The condition $\cos x > 0$ holds when $x$ lies in intervals of the form $(2n\pi - \frac{\pi}{2}, 2n\pi + \frac{\pi}{2})$ for any integer $n$.
For example, one such interval is $(-\frac{\pi}{2}, \frac{\pi}{2})$.
The domain of $F(x)$ is $D = \bigcup_{n \in \mathbb{Z}} (2n\pi - \frac{\pi}{2}, 2n\pi + \frac{\pi}{2})$.

2. Check for Even/Odd nature:

A function $F(x)$ is even if $F(-x) = F(x)$ for all $x$ in its domain.
A function $F(x)$ is odd if $F(-x) = -F(x)$ for all $x$ in its domain.

First, we check if the domain is symmetric about the origin. If $x \in D$, then $\cos x > 0$. Since $\cos(-x) = \cos x$, it implies $\cos(-x) > 0$. Thus, if $x \in D$, then $-x \in D$. The domain is symmetric about the origin, so the function can be even or odd.

Now, let's evaluate $F(-x)$:
$F(-x) = \ln(\cos(-x))$

Since the cosine function is an even function, $\cos(-x) = \cos x$.
Therefore,
$F(-x) = \ln(\cos x)$

This is equal to $F(x)$.
Since $F(-x) = F(x)$, the function $F(x) = \ln(\cos x)$ is an even function.

3. Check for Periodic nature:

A function $F(x)$ is periodic if there exists a positive constant $T$ such that $F(x+T) = F(x)$ for all $x$ in its domain. The smallest such $T$ is called the period.

We know that the cosine function is periodic with a period of $2\pi$, meaning $\cos(x+2\pi) = \cos x$.

Let's evaluate $F(x+2\pi)$:
$F(x+2\pi) = \ln(\cos(x+2\pi))$

Since $\cos(x+2\pi) = \cos x$,
$F(x+2\pi) = \ln(\cos x)$

This is equal to $F(x)$.
Since $F(x+2\pi) = F(x)$, the function $F(x) = \ln(\cos x)$ is a periodic function with a period of $2\pi$.

Conclusion:

The function $F(x) = \ln(\cos x)$ is both an even function and a periodic function.