Question

Формулы сумы и разницы 4) ctg (45° -a) - ctg (45° +a) 5) cos 16° - cos 36° 6) sin 28° + sin 12° 7) cos 3a + cos 5a 8) ctg 55° - ctg 15°

Answer 1

$4)~\rm{ctg}\,(45^\circ-\alpha)-\rm{ctg}\,(45^\circ+\alpha)=\dfrac{\sin(45^\circ+\alpha-45^\circ+\alpha)}{\sin(45^\circ-\alpha)\sin(45^\circ+\alpha)}=\\ \\ =\dfrac{\sin2\alpha}{\sin(45^\circ-\alpha)\sin(45^\circ+\alpha)}$

$5)~ \cos16^\circ-\cos36^\circ=2\sin\dfrac{16^\circ+36^\circ}{2}\sin\dfrac{36^\circ-16^\circ}{2}=2\sin26^\circ\sin10^\circ$

$6)~ \sin28^\circ+\sin12^\circ=2\sin\dfrac{28^\circ+12^\circ}{2}\cos\dfrac{28^\circ-12^\circ}{2}=2\sin20^\circ\cos8^\circ$

$7)~ \cos3\alpha+\cos5\alpha=2\cos\dfrac{3\alpha+5\alpha}{2}\cos\dfrac{3\alpha-5\alpha}{2}=2\cos4\alpha\cos\alpha$

$8)~ \rm{ctg}\,55^\circ-\rm{ctg}\,15^\circ=\dfrac{\sin(15^\circ-55^\circ)}{\sin55^\circ\sin15^\circ}=-\dfrac{\sin40^\circ}{\sin55^\circ\sin15^\circ}$