The problem in the image asks to determine the trigonometric values without using a table or calculator. Let's go through the first problem, which involves simplifying the expression for $\sin 70^\circ \sin 20^\circ$.

The steps provided in the image are based on a product-to-sum trigonometric identity:

$\sin A \sin B = \frac{1}{2} [\cos(A - B) - \cos(A + B)]$

For $A = 70^\circ$ and $B = 20^\circ$, applying this identity gives:

$\sin 70^\circ \sin 20^\circ = \frac{1}{2} [\cos(70^\circ - 20^\circ) - \cos(70^\circ + 20^\circ)]$ $= \frac{1}{2} [\cos 50^\circ - \cos 90^\circ]$

Since $\cos 90^\circ = 0$, this simplifies to:

$\sin 70^\circ \sin 20^\circ = \frac{1}{2} \cos 50^\circ$

Thus, the value of $\sin 70^\circ \sin 20^\circ$ is:

$\sin 70^\circ \sin 20^\circ = \frac{1}{2} \cos 50^\circ$

Would you like to go over more details or have any questions?

Here are 5 follow-up questions for further exploration:

1. How would you use the product-to-sum identity to simplify $\cos 75^\circ \cos 15^\circ$?
2. What is the value of $\cos 90^\circ$ and why is it important in trigonometric simplifications?
3. Can you explain the difference between sum-to-product and product-to-sum identities?
4. How would you calculate $\sin(75^\circ) + \sin(15^\circ)$ using sum identities?
5. Why does the product-to-sum identity work for simplifying trigonometric expressions?

Tip: Memorizing key trigonometric identities, like product-to-sum formulas, helps solve complex expressions quickly without using calculators.