Let's break down and solve **Question 8** and the ones that follow.

### **Question 8: Trigonometry**

You are asked to analyze a sinusoidal function based on a provided sketch (which isn't visible here, so we'll focus on general steps) and prove a trigonometric identity.

#### **Part a**: Analyzing the Sinusoidal Function

For the sinusoidal function, you're required to identify the following characteristics:
- **Amplitude**
- **Period**
- **Phase Shift**
- **Axis of Symmetry**

You would typically find these by examining the equation of the sinusoidal function, which might look like:
\[ y = a \cdot \sin(b(x - c)) + d \]
Or:
\[ y = a \cdot \cos(b(x - c)) + d \]

Here’s what each parameter represents:
- **Amplitude** \((a)\): The vertical stretch/shrink of the function. The amplitude is the absolute value of \(a\).
- **Period** \((T)\): Calculated using the formula \( T = \frac{2\pi}{|b|} \).
- **Phase Shift** \((c)\): The horizontal shift of the graph, determined by the value \(c\).
- **Axis of Symmetry** \((d)\): The horizontal line around which the function oscillates, given by the value \(d\).

#### **Part b**: Proving the Identity

You need to prove the following trigonometric identity:
\[
(\sec x + \csc x)(\sin x + \cos x) = \sec x \csc x + 2
\]

**Step-by-Step Proof:**

1. **Expand the Left-Hand Side (LHS):**

   Start by expanding the LHS:
   \[
   (\sec x + \csc x)(\sin x + \cos x) = \sec x \sin x + \sec x \cos x + \csc x \sin x + \csc x \cos x
   \]

2. **Simplify Each Term:**
   \[
   \sec x \sin x = \frac{1}{\cos x} \cdot \sin x = \frac{\sin x}{\cos x} = \tan x
   \]
   \[
   \sec x \cos x = \frac{1}{\cos x} \cdot \cos x = 1
   \]
   \[
   \csc x \sin x = \frac{1}{\sin x} \cdot \sin x = 1
   \]
   \[
   \csc x \cos x = \frac{1}{\sin x} \cdot \cos x = \cot x
   \]

   So, the LHS becomes:
   \[
   \tan x + 1 + 1 + \cot x = \tan x + \cot x + 2
   \]

3. **Express in Terms of \(\sec x \csc x\):**
  

Explore how to analyze a sinusoidal function and prove a trigonometric identity involving secant, cosecant, sine, and cosine functions. This problem is suitable for high school students learning trigonometry.

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