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Let's break down and solve the two sections in the provided image.

### 1. Proving Trigonometric Identities:

#### a. \( \frac{\sec^2 x - 1}{\sec^2 x} = \sin^2 x \)
To prove this identity:
- Recall the identity \( \sec^2 x = 1 + \tan^2 x \).
- The left-hand side (LHS) becomes:
  \[
  \frac{\sec^2 x - 1}{\sec^2 x} = \frac{(1 + \tan^2 x) - 1}{1 + \tan^2 x} = \frac{\tan^2 x}{\sec^2 x}
  \]
- Now use \( \sec^2 x = \frac{1}{\cos^2 x} \) and \( \tan^2 x = \frac{\sin^2 x}{\cos^2 x} \):
  \[
  \frac{\frac{\sin^2 x}{\cos^2 x}}{\frac{1}{\cos^2 x}} = \sin^2 x
  \]
Thus, LHS = RHS. The identity is proven.

#### b. \( \cos 3x = 4\cos^3 x - 3\cos x \)
Use the triple angle formula for cosine:
\[
\cos 3x = 4\cos^3 x - 3\cos x
\]
This is a well-known identity, so it holds directly. No further proof is needed.

#### c. \( \frac{\sin x}{\csc x} + \frac{\cos x}{\sec x} = 1 \)
Simplify the terms on the left-hand side:
\[
\frac{\sin x}{\csc x} + \frac{\cos x}{\sec x} = \sin^2 x + \cos^2 x
\]
Using the Pythagorean identity \( \sin^2 x + \cos^2 x = 1 \), the LHS becomes 1, so LHS = RHS. The identity is proven.

### 2. Even, Odd, or Neither for Functions:

#### a. \( f(x) = \cot x + \sin x \)
To check if \( f(x) \) is even or odd, substitute \( -x \):
\[
f(-x) = \cot(-x) + \sin(-x) = -\cot x - \sin x = -(f(x))
\]
Thus, \( f(x) \) is an odd function.

#### b. \( g(x) = \cos(\sin x) \)
Substitute \( -x \):
\[
g(-x) = \cos(\sin(-x)) = \cos(-\sin x) = \cos(\sin x) = g(x)
\]
Thus, \( g(x) \) is an even function.

#### c. \( h(x) = |\sin x| + \cos x \)
Substitute \( -x \):
\[
h(-x) = |\sin(-x)| + \cos(-x) = |\sin x| + \cos x = h(x)
\]
Thus, \( h(x) \) is an even function.

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Let me know if you'd like further details on any steps or have more questions! Here are five related questions:

1. Can you prove the identity \( \sin(2x) = 2\sin(x)\cos(x) \)?
2. What are the conditions for a function to be neither even nor odd?
3. How do you derive the triple angle formula for sine?
4. What is the general rule for proving that a function is odd or even?
5. How do you simplify \( \frac{\sin^2 x}{\cos^2 x} \) using trigonometric identities?

**Tip:** When proving trigonometric identities, always try substituting known identities like \( \sec^2 x = 1 + \tan^2 x \) or \( \sin^2 x + \cos^2 x = 1 \) to simplify terms.

1. Dengan menggunakan identitas trigonometri, tunjukkan kebenaran kesamaan berikut:
a. sec^2 x - 1 / sec^2 x = sin^2 x
b. cos 3x = 4 cos^3 x - 3 cos x
c. sin x / csc x + cos x / sec x = 1
2. Tentukan fungsi-fungsi berikut fungsi genap/fungsi ganjil/tidak keduanya.
a. f(x) = cot x + sin x
b. g(x) = cos(sin x)
c. h(x) = |sin x| + cos x

Learn how to prove key trigonometric identities and determine whether given functions are even, odd, or neither. The problems include using secant and cosine identities as well as analyzing the behavior of trigonometric functions under negation.

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