To solve these exercises, we’ll apply trigonometric addition and subtraction formulas to rewrite each expression as a single trigonometric function. I'll go through each one in order.

Problem 15

Expression: $\sin 18^\circ \cos 27^\circ + \cos 18^\circ \sin 27^\circ$

This is in the form of the sine addition formula: $\sin A \cos B + \cos A \sin B = \sin(A + B)$ Thus, $\sin 18^\circ \cos 27^\circ + \cos 18^\circ \sin 27^\circ = \sin(18^\circ + 27^\circ) = \sin 45^\circ$ Since $\sin 45^\circ = \frac{\sqrt{2}}{2}$, the exact value is: $\frac{\sqrt{2}}{2}$

Problem 16

Expression: $\cos 10^\circ \cos 80^\circ - \sin 10^\circ \sin 80^\circ$

This matches the cosine subtraction formula: $\cos A \cos B - \sin A \sin B = \cos(A + B)$ Therefore, $\cos 10^\circ \cos 80^\circ - \sin 10^\circ \sin 80^\circ = \cos(10^\circ + 80^\circ) = \cos 90^\circ$ Since $\cos 90^\circ = 0$, the exact value is: $0$

Problem 17

Expression: $\cos \frac{3\pi}{7} \cos \frac{2\pi}{21} + \sin \frac{3\pi}{7} \sin \frac{2\pi}{21}$

This is in the form of the cosine addition formula: $\cos A \cos B + \sin A \sin B = \cos(A - B)$ Thus, $\cos \frac{3\pi}{7} \cos \frac{2\pi}{21} + \sin \frac{3\pi}{7} \sin \frac{2\pi}{21} = \cos\left(\frac{3\pi}{7} - \frac{2\pi}{21}\right)$ Now, finding a common denominator for $\frac{3\pi}{7}$ and $\frac{2\pi}{21}$: $\frac{3\pi}{7} = \frac{9\pi}{21}, \quad \frac{2\pi}{21} = \frac{2\pi}{21}$ So, $\frac{3\pi}{7} - \frac{2\pi}{21} = \frac{9\pi - 2\pi}{21} = \frac{7\pi}{21} = \frac{\pi}{3}$ Therefore, $\cos \frac{3\pi}{7} \cos \frac{2\pi}{21} + \sin \frac{3\pi}{7} \sin \frac{2\pi}{21} = \cos \frac{\pi}{3}$ Since $\cos \frac{\pi}{3} = \frac{1}{2}$, the exact value is: $\frac{1}{2}$

Problem 18

Expression: $\frac{\tan \frac{\pi}{18} + \tan \frac{\pi}{9}}{1 - \tan \frac{\pi}{18} \tan \frac{\pi}{9}}$

This matches the tangent addition formula: $\frac{\tan A + \tan B}{1 - \tan A \tan B} = \tan(A + B)$ Thus, $\frac{\tan \frac{\pi}{18} + \tan \frac{\pi}{9}}{1 - \tan \frac{\pi}{18} \tan \frac{\pi}{9}} = \tan\left(\frac{\pi}{18} + \frac{\pi}{9}\right)$ Finding a common denominator for $\frac{\pi}{18}$ and $\frac{\pi}{9}$: $\frac{\pi}{18} + \frac{\pi}{9} = \frac{\pi}{18} + \frac{2\pi}{18} = \frac{3\pi}{18} = \frac{\pi}{6}$ Therefore, $\frac{\tan \frac{\pi}{18} + \tan \frac{\pi}{9}}{1 - \tan \frac{\pi}{18} \tan \frac{\pi}{9}} = \tan \frac{\pi}{6}$ Since $\tan \frac{\pi}{6} = \frac{1}{\sqrt{3}}$, the exact value is: $\frac{1}{\sqrt{3}}$

Would you like further details on any step, or do you have additional questions?

Here are some related questions you may find helpful:

1. How do we derive the sine addition formula?
2. Can we apply the tangent addition formula to larger expressions?
3. What are other methods to simplify trigonometric expressions?
4. How do these formulas apply in real-world situations?
5. Why is it useful to know exact values for common angles?

Tip: Memorizing the exact values for common angles like $30^\circ$, $45^\circ$, and $60^\circ$ can make trigonometric simplifications much easier.