Let’s work through each of these expressions and simplify them into single trigonometric ratios:

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

Using the double-angle identity: $\sin 2A = 2 \sin A \cos A$ Substituting $A = 5x$, we get: $2 \sin 5x \cos 5x = \sin 10x$

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

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

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

Using the cosine double-angle identity: $\cos 2A = 1 - 2 \sin^2 A$ Substituting $A = 3x$, we get: $1 - 2 \sin^2 3x = \cos 6x$

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

Using the tangent double-angle identity: $\tan 2A = \frac{2 \tan A}{1 - \tan^2 A}$ Substituting $A = 4x$, we get: $\frac{2 \tan 4x}{1 - \tan^2 4x} = \tan 8x$

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

Using the double-angle identity $\sin 2A = 2 \sin A \cos A$: $4 \sin \theta \cos \theta = 2 \cdot 2 \sin \theta \cos \theta = 2 \sin 2\theta$

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

Using the cosine double-angle identity: $\cos 2A = 2 \cos^2 A - 1$ Substituting $A = \frac{\theta}{2}$, we get: $2 \cos^2 \frac{\theta}{2} - 1 = \cos \theta$

Final Simplified Expressions:

a) $\sin 10x$
b) $\cos 2\theta$
c) $\cos 6x$
d) $\tan 8x$
e) $2 \sin 2\theta$
f) $\cos \theta$

Would you like further clarifications or visual explanations?

Here are 5 related questions to deepen your understanding:

1. What are the derivations of the trigonometric double-angle identities?
2. How do you use product-to-sum identities to simplify similar expressions?
3. Can you generalize these concepts for triple-angle identities?
4. How do these transformations help in solving trigonometric equations?
5. What are some common mistakes when applying these identities?

Tip: Always verify which identity applies best by matching the form of the equation with standard trigonometric identities.