The image contains trigonometric equations and expressions to be solved or simplified. I'll analyze the mathematical content and solve step by step:

1. Equation 1:

$\sin \alpha \cos \alpha = \frac{\sqrt{2}}{8}$

Solution:

Using the identity $\sin \alpha \cos \alpha = \frac{1}{2} \sin(2\alpha)$: $\frac{1}{2} \sin(2\alpha) = \frac{\sqrt{2}}{8}$ $\sin(2\alpha) = \frac{\sqrt{2}}{4}$ Finding the general solution for $\sin(2\alpha) = \frac{\sqrt{2}}{4}$: $2\alpha = \arcsin\left(\frac{\sqrt{2}}{4}\right) + 2n\pi \quad \text{or} \quad 2\alpha = \pi - \arcsin\left(\frac{\sqrt{2}}{4}\right) + 2n\pi$ $\alpha = \frac{1}{2} \arcsin\left(\frac{\sqrt{2}}{4}\right) + n\pi \quad \text{or} \quad \alpha = \frac{\pi}{2} - \frac{1}{2} \arcsin\left(\frac{\sqrt{2}}{4}\right) + n\pi$

2. Equation 2:

$2 \sin^2(3\theta) + \sin^2(6\theta) = 2$

Solution:

The maximum value of $\sin^2(x)$ is 1. To satisfy the equation, both $\sin^2(3\theta)$ and $\sin^2(6\theta)$ must be equal to 1. Thus: $\sin^2(3\theta) = 1 \quad \text{and} \quad \sin^2(6\theta) = 1$ This implies: $3\theta = \frac{\pi}{2} + n\pi \quad \text{and} \quad 6\theta = \frac{\pi}{2} + m\pi$ For $\sin^2(6\theta)$, solving gives: $\theta = \frac{\pi}{6} + \frac{n\pi}{3} \quad \text{for integers } n.$

3. Equation 3:

$\tan(\theta) + \tan(2\theta) + \tan(3\theta) = 0$

Solution:

Using the tangent sum formula and properties: Let $x = \tan(\theta), y = \tan(2\theta), z = \tan(3\theta)$. Then: $x + y + z = 0$ This is a transcendental equation, requiring numeric or analytic techniques for specific solutions.

4. Expression to Simplify:

$P = \cos^2(A) + \cos^2(B) + \cos^2(C) - 2 \cos(A) \cos(B) \cos(C) - 1$

Solution:

Using the Pythagorean identity $\cos^2(x) = 1 - \sin^2(x)$, rewrite $P$ to simplify. Substituting further will depend on the angle sum constraints of $A, B,$ and $C$.

Would you like step-by-step details or numerical solutions for a specific part? Let me know!

Further Questions:

1. How can trigonometric identities simplify complex expressions like $P$?
2. What are the properties of solutions to $\sin(x) = k$?
3. How do periodic solutions appear in tangent equations like Equation 3?
4. How does $\arcsin(x)$ behave for values beyond the principal range?
5. Can $P$ simplify further under additional constraints on $A, B, C$?

Tip:

Always visualize periodic functions like $\sin(x), \cos(x), \tan(x)$ on their respective intervals to better understand solutions!