Solve the equation 2(x) + 2(x) = 1 for all x. Prove the identity (2x) = 2(x) − 2(x). Prove that (A−B) = ((A) − (B)) / (1 + (A)(B)).

Here are the solutions.

3. Solve the equation $\sin^2(x) + \cos^2(x) = 1$ for all x.

This is the Pythagorean trigonometric identity.
The equation is true for all real values of $x$.
The final answer is $\boxed{\text{All real numbers}}$.

4. Prove the identity $\cos(2x) = \cos^2(x) - \sin^2(x)$.

Start with the sum formula for cosine:

$$\cos(A+B) = \cos(A)\cos(B) - \sin(A)\sin(B)$$

Let $A=x$ and $B=x$:

$$\cos(x+x) = \cos(x)\cos(x) - \sin(x)\sin(x)$$ $$\cos(2x) = \cos^2(x) - \sin^2(x)$$

The identity is proven.

5. Prove that $\tan(A-B) = (\tan(A) - \tan(B)) / (1 + \tan(A)\tan(B))$.

Start with the definition of tangent and the difference formulas for sine and cosine:

$$\tan(A-B) = \frac{\sin(A-B)}{\cos(A-B)}$$ $$= \frac{\sin(A)\cos(B) - \cos(A)\sin(B)}{\cos(A)\cos(B) + \sin(A)\sin(B)}$$

Divide the numerator and denominator by $\cos(A)\cos(B)$:

$$= \frac{\frac{\sin(A)\cos(B)}{\cos(A)\cos(B)} - \frac{\cos(A)\sin(B)}{\cos(A)\cos(B)}}{\frac{\cos(A)\cos(B)}{\cos(A)\cos(B)} + \frac{\sin(A)\sin(B)}{\cos(A)\cos(B)}}$$ $$= \frac{\frac{\sin(A)}{\cos(A)} - \frac{\sin(B)}{\cos(B)}}{1 + \frac{\sin(A)}{\cos(A)}\frac{\sin(B)}{\cos(B)}}$$ $$= \frac{\tan(A) - \tan(B)}{1 + \tan(A)\tan(B)}$$

The identity is proven.