9. Вычислите: а) [cos(π/15) cos(4π/15) − sin(4π/15) sin(π/15)]/[cos(3π/10) sin(0,2π) + sin(0,3π) cos(0,2π)]; б) [tg(2π/3) − tg(5π/12)]/[1 + tg(2π/3) tg(5π/12)]; в) [tg(π/10) + tg(3π/20)]/[1 − tg(π/10) tg(3π/20)]; г) [sin(5π/18) cos(π/9) − sin(π/9) cos(5π/18)]/[sin(5π/12) sin(7π/12) − cos(5π/12) cos(7π/12)].

а) В числителе применяем формулу косинуса суммы:

cos(π/15) cos(4π/15) − sin(4π/15) sin(π/15) = cos(5π/15) = cos(π/3) = 1/2.

В знаменателе:

cos(3π/10) sin(π/5) + sin(3π/10) cos(π/5) = sin(3π/10 + π/5) = sin(π/2) = 1.

Следовательно, результат равен \(\dfrac{1/2}{1}=\dfrac12\).

б) По формуле тангенса разности:

\(\dfrac{ g\frac{2\pi}{3}- g\frac{5\pi}{12}}{1+ g\frac{2\pi}{3} g\frac{5\pi}{12}}= g\left(\frac{2\pi}{3}-\frac{5\pi}{12}\right)= g\frac{\pi}{4}=1\).

в) По формуле тангенса суммы:

\(\dfrac{ g\frac{\pi}{10}+ g\frac{3\pi}{20}}{1- g\frac{\pi}{10} g\frac{3\pi}{20}}= g\left(\frac{\pi}{10}+\frac{3\pi}{20}\right)= g\frac{\pi}{4}=1\).

г) В числителе:

sin(5π/18) cos(π/9) − sin(π/9) cos(5π/18) = sin(5π/18 − π/9) = sin(π/6) = 1/2.

В знаменателе:

sin(5π/12) sin(7π/12) − cos(5π/12) cos(7π/12) = −cos(5π/12 + 7π/12) = −cos π = 1.

Итак, результат равен \(\dfrac{1/2}{1}=\dfrac12\).

Ответ: а) 1/2; б) 1; в) 1; г) 1/2.