Let's solve the problems step-by-step based on the given image.

11. Verify the cofunction identities using compound angle formulas:

a) $\sin x = \cos\left(\frac{\pi}{2} - x\right)$

• Using the identity for cosine of a difference: $\cos\left(\frac{\pi}{2} - x\right) = \sin x$ This directly verifies the identity.

b) $\cos x = \sin\left(\frac{\pi}{2} - x\right)$

• Using the identity for sine of a difference: $\sin\left(\frac{\pi}{2} - x\right) = \cos x$ This directly verifies the identity.

12. Simplify each expression:

a) $\sin(\pi + x) + \sin(\pi - x)$

• Using the sine addition and subtraction formulas: $\sin(\pi + x) = -\sin x \quad \text{and} \quad \sin(\pi - x) = \sin x$
• Substituting: $\sin(\pi + x) + \sin(\pi - x) = -\sin x + \sin x = 0$

b) $\cos\left(x + \frac{\pi}{3}\right) - \sin\left(x + \frac{\pi}{6}\right)$

• Using the formulas for cosine and sine of sums: $\cos\left(x + \frac{\pi}{3}\right) = \cos x \cos\frac{\pi}{3} - \sin x \sin\frac{\pi}{3}$ $\sin\left(x + \frac{\pi}{6}\right) = \sin x \cos\frac{\pi}{6} + \cos x \sin\frac{\pi}{6}$
• Substituting values: $\cos\frac{\pi}{3} = \frac{1}{2}, \, \sin\frac{\pi}{3} = \frac{\sqrt{3}}{2}, \, \cos\frac{\pi}{6} = \frac{\sqrt{3}}{2}, \, \sin\frac{\pi}{6} = \frac{1}{2}$
• Expanding: $\cos\left(x + \frac{\pi}{3}\right) = \cos x \cdot \frac{1}{2} - \sin x \cdot \frac{\sqrt{3}}{2}$ $\sin\left(x + \frac{\pi}{6}\right) = \sin x \cdot \frac{\sqrt{3}}{2} + \cos x \cdot \frac{1}{2}$
• Subtracting: $\cos\left(x + \frac{\pi}{3}\right) - \sin\left(x + \frac{\pi}{6}\right) = \cos x \cdot \frac{1}{2} - \sin x \cdot \frac{\sqrt{3}}{2} - \left(\sin x \cdot \frac{\sqrt{3}}{2} + \cos x \cdot \frac{1}{2}\right)$ $= 0$

13. Simplify:

$\frac{\sin(f + g) + \sin(f - g)}{\cos(f + g) + \cos(f - g)}$

• Using sum-to-product identities: $\sin(f + g) + \sin(f - g) = 2 \sin f \cos g$ $\cos(f + g) + \cos(f - g) = 2 \cos f \cos g$
• Substituting: $\frac{\sin(f + g) + \sin(f - g)}{\cos(f + g) + \cos(f - g)} = \frac{2 \sin f \cos g}{2 \cos f \cos g}$
• Simplify: $\frac{\sin f}{\cos f} = \tan f$
• Final answer: $\tan f$

Do you want further clarification or detailed explanations for any of these? Let me know!

Related Questions:

1. How do the sum-to-product identities apply to trigonometric expressions?
2. Can you derive the sine and cosine addition formulas step-by-step?
3. Why do the cofunction identities hold for all $x$?
4. How can you simplify similar trigonometric ratios without a calculator?
5. What are the common mistakes to avoid when applying these identities?

Tip:

Always remember to verify your results by substituting specific values for the variables to ensure consistency!