Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be the function given by $f(x) = \cos(\tan^{-1} x)$. Which one of the following statements is TRUE?
Show Hint
An alternate graphical way is to recognize that $f(x) = \frac{1}{\sqrt{1+x^2}}$ is an even function symmetric about the y-axis, with its maximum at $x=0$.
As $x$ moves away from 0 in the positive direction, the denominator increases, which means $f(x)$ must decrease for $x > 0$.
Step 1 : Understanding the Question:
The question asks us to identify the interval on which the composite trigonometric function $f(x) = \cos(\tan^{-1} x)$ is decreasing. Step 2 : Key Formulas and Approach:
We will first simplify the composite function $f(x)$ into an algebraic form using trigonometric identities.
Then, we will compute the first derivative $f'(x)$ and determine its sign on different intervals:
• If $f'(x) < 0$, the function is decreasing.
• If $f'(x) > 0$, the function is increasing. Step 3 : Detailed Explanation:
Let us simplify $f(x) = \cos(\tan^{-1} x)$:
Let $\theta = \tan^{-1} x \implies \tan \theta = x$.
Since the range of $\tan^{-1} x$ is $(-\frac{\pi}{2}, \frac{\pi}{2})$, $\cos \theta$ is always positive.
Using the right-angled triangle relationship:
\[ \cos \theta = \frac{1}{\sec \theta} = \frac{1}{\sqrt{1 + \tan^2 \theta}} = \frac{1}{\sqrt{1 + x^2}} \]
Thus, we can write $f(x)$ as:
\[ f(x) = (1 + x^2)^{-1/2} \]
Now, let us differentiate $f(x)$ with respect to $x$ using the chain rule:
\[ f'(x) = -\frac{1}{2}(1 + x^2)^{-3/2} \cdot \frac{d}{dx}(1 + x^2) \]
\[ f'(x) = -\frac{1}{2}(1 + x^2)^{-3/2} \cdot (2x) \]
\[ f'(x) = -\frac{x}{(1 + x^2)^{3/2}} \]
Let us analyze the sign of $f'(x)$:
• Since the denominator $(1 + x^2)^{3/2}$ is always positive for all real $x$, the sign of $f'(x)$ depends solely on the numerator $-x$.
• Case 1: $x > 0$
Here, $-x < 0 \implies f'(x) < 0$.
Thus, $f(x)$ is strictly decreasing for $x > 0$.
• Case 2: $x < 0$
Here, $-x > 0 \implies f'(x) > 0$.
Thus, $f(x)$ is strictly increasing for $x < 0$. Step 4 : Final Answer:
The function $f(x)$ is decreasing for $x > 0$.
This matches Option (A).
