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.