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Let's go through the questions shown in the image.

### 1. Perform the indicated operations and, if possible, simplify your answers. Leave all answers in terms of \(\sin \theta\) and \(\cos \theta\).

#### a) \(\sec \theta \cot \theta\)

Start by expressing \(\sec \theta\) and \(\cot \theta\) in terms of \(\sin \theta\) and \(\cos \theta\):
\[
\sec \theta = \frac{1}{\cos \theta}, \quad \cot \theta = \frac{\cos \theta}{\sin \theta}
\]
Now multiply them together:
\[
\sec \theta \cot \theta = \frac{1}{\cos \theta} \cdot \frac{\cos \theta}{\sin \theta} = \frac{1}{\sin \theta}
\]
So, the simplified form is:
\[
\sec \theta \cot \theta = \csc \theta
\]

#### b) \(\tan^2 \theta - \sec^2 \theta\)

Use the trigonometric identity \(\sec^2 \theta - \tan^2 \theta = 1\), then rearrange it:
\[
\tan^2 \theta - \sec^2 \theta = -1
\]

#### c) \(\csc \theta - \sin \theta\)

Express \(\csc \theta\) in terms of \(\sin \theta\):
\[
\csc \theta = \frac{1}{\sin \theta}
\]
Now subtract \(\sin \theta\):
\[
\csc \theta - \sin \theta = \frac{1}{\sin \theta} - \sin \theta
\]
Combine the terms:
\[
= \frac{1 - \sin^2 \theta}{\sin \theta}
\]
Using the identity \(1 - \sin^2 \theta = \cos^2 \theta\):
\[
\csc \theta - \sin \theta = \frac{\cos^2 \theta}{\sin \theta}
\]

### 2. Using sum and difference formulas, show that:

#### a) \(\sin(2x) \sin(3x) + \cos(2x) \cos(3x) = \cos(x)\)

This is a direct application of the sum formula for cosine:
\[
\cos(A - B) = \cos A \cos B + \sin A \sin B
\]
Set \(A = 3x\) and \(B = 2x\):
\[
\cos(3x - 2x) = \cos(3x) \cos(2x) + \sin(3x) \sin(2x)
\]
This simplifies to:
\[
\cos(x) = \cos(2x) \cos(3x) + \sin(2x) \sin(3x)
\]
Thus, the expression holds.

#### b) \(\cos(\pi - x) \sin(x) + \sin(\pi - x) \cos(x) = 0\)

Using the known identities:
\[
\cos(\pi - x) = -\cos(x), \quad \sin(\pi - x) = \sin(x)
\]
Substitute these into the expression:
\[
-\cos(x) \sin(x) + \sin(x) \cos(x)
\]
This simplifies to:
\[
-\cos(x) \sin(x) + \cos(x) \sin(x) = 0
\]

Would you like further details or have any questions?

### Related questions:
1. How do the reciprocal identities (e.g., \(\sec \theta\), \(\csc \theta\)) relate to the basic trigonometric functions?
2. Can you prove \(\sec^2 \theta - \tan^2 \theta = 1\) using the Pythagorean identity?
3. How would the expressions change if the problem used \(\tan\) and \(\cot\) instead of \(\sec\) and \(\csc\)?
4. What are some applications of sum and difference formulas in solving real-world problems?
5. Can you apply the same process to derive the difference formula for sine?

### Tip:
Always try simplifying trigonometric expressions using identities like the Pythagorean identities, reciprocal identities, and sum/difference formulas.

1. Perform the indicated operations and if possible, simplify your answers. Leave all answers in terms of sin(θ) and cos(θ). 
a) sec(θ) cot(θ) 
b) tan^2(θ) - sec^2(θ) 
c) csc(θ) - sin(θ) 

2. Using sum and difference formulas, show that: 
a) sin(2x) sin(3x) + cos(2x) cos(3x) = cos(x) 
b) cos(π - x) sin(x) + sin(π - x) cos(x) = 0 (note that π rad = 180 degrees).

Learn how to simplify trigonometric expressions involving sec(θ), cot(θ), and tan(θ), and solve trigonometric equations using sum and difference formulas. This guide is perfect for high school students in grades 10-12, providing step-by-step explanations.

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