Questions: p(x) = (csc x tan x + e^x) / (x^2 sec x)

Transcript text: $p(x)=\frac{\csc x \tan x+e^{x}}{x^{2} \sec x}$

Solution

Solution Steps

To solve for $ p(x) $, we need to simplify the given expression. We will break down the trigonometric and exponential functions and then combine them according to the given formula.

Simplify the trigonometric functions: $\csc x$, $\tan x$, and $\sec x$.
Substitute these simplified forms into the given expression.
Perform the division and simplification.

Step 1: Define the Expression

We start with the expression for $ p(x) $: \[ p(x) = \frac{\csc x \tan x + e^{x}}{x^{2} \sec x} \]

Step 2: Simplify Trigonometric Functions

Using the identities: \[ \csc x = \frac{1}{\sin x}, \quad \tan x = \frac{\sin x}{\cos x}, \quad \sec x = \frac{1}{\cos x} \] we can rewrite the expression as: \[ p(x) = \frac{\frac{1}{\sin x} \cdot \frac{\sin x}{\cos x} + e^{x}}{x^{2} \cdot \frac{1}{\cos x}} = \frac{\frac{1}{\cos x} + e^{x}}{x^{2} \cdot \frac{1}{\cos x}} \]

Step 3: Combine and Simplify

This simplifies to: \[ p(x) = \frac{e^{x} + 1}{x^{2}} \]