The equations provided in the image appear to involve trigonometric identities. Let me simplify each expression:

a) $2 \sin(5x) \cos(5x)$

Using the double angle identity: $\sin(2\theta) = 2 \sin(\theta) \cos(\theta)$ This simplifies to: $\sin(10x)$

b) $\cos^2\theta - \sin^2\theta$

Using the cosine double angle identity: $\cos(2\theta) = \cos^2\theta - \sin^2\theta$ This simplifies to: $\cos(2\theta)$

c) $1 - 2\sin^2(3x)$

Using the cosine double angle identity: $\cos(2\theta) = 1 - 2\sin^2\theta$ This simplifies to: $\cos(6x)$

d) $\frac{2\tan(4x)}{1 - \tan^2(4x)}$

Using the tangent double angle identity: $\tan(2\theta) = \frac{2\tan(\theta)}{1 - \tan^2(\theta)}$ This simplifies to: $\tan(8x)$

e) $4\sin\theta \cos\theta$

Using the double angle identity: $\sin(2\theta) = 2 \sin(\theta) \cos(\theta)$ This simplifies to: $2\sin(2\theta)$

f) $2\cos^2\left(\frac{\theta}{2}\right) - 1$

Using the cosine double angle identity: $\cos(2\theta) = 2\cos^2(\theta) - 1$ This simplifies to: $\cos(\theta)$

If you would like detailed step-by-step explanations for any specific identity or calculation, let me know!

Related Questions:

1. Can you derive the double-angle formula for tangent starting from sine and cosine identities?
2. What is the relationship between the tangent and sine/cosine double angle identities?
3. How do the simplifications change when using triple-angle or half-angle identities?
4. Can these expressions be represented as Euler's formula ($e^{i\theta}$)? How?
5. What are some real-life applications of trigonometric identities like these?

Tip: Memorize fundamental trigonometric identities like double-angle and half-angle formulas—they are frequently used in simplifying complex problems.