1. Упростите выражение: а) sin 10a 2sin²5a б) cos6a cos 3a+sin 3aв)(sin a +cos a)² ...

$1)\; a)\; \frac{sin10a}{2sin^25a}=\frac{2sin5a\cdot cos5a}{2sin^25a}=\frac{cos5a}{sin5a}=ctg5a\\\\b)\; \frac{cos6a}{cos3a+sin3a}=\frac{cos^23a-sin^23a}{cos3a+sin3a}=\frac{(cos3a-sin3a)(cos3a+sin3a)}{cos3a+sin3a}=\\\\=cos3a-sin3a=cos3a-cos(\frac{\pi}{2}-3a)=\\\\--2sin\frac{\pi}{4}sin((3a-\frac{\pi}{4})=-\sqrt2sin(3a-\frac{\pi}{4})\\\\c)\; \frac{(sina+cosa)^2}{1+sin2a}=\frac{(sin^2a+cos^2a)+2sinacosa}{1+sin2a}=\frac{1+sin2a}{1+sin2a}=1\\\\d)\; 2sin^2\frac{a}{2}+cosa=2sin^2\frac{a}{2}+(1-2sin^2\frac{a}{2})=1$

$2)\; cos^2\frac{\pi}{12}-sin^2\frac{\pi}{12}=cos\frac{\pi}{6}=\frac{\sqrt3}{2}\\\\4sin\frac{\pi}{8}\cdot cos\frac{\pi}{8}=2sin\frac{\pi}{4}=2\frac{\sqrt2}{2}=\sqrt2\\\\2cos^275^0-1=2cos^275^0-(sin^275^0+cos^275^0)=cos^275^0-sin^275^0=\\\\=cos150^0=cos(180^0-30^0)=-cos30^0=-\frac{\sqrt3}{2}\\\\3)\; sinxcos(-x)=-\frac{\sqrt{3}}{4}\\\\cos(-x)=cosx\\\\sinx\cdot cosx=-\frac{\sqrt3}{2}\\\\\frac{1}{2}sin2x=-\frac{\sqrt3}{4}\\\\sin2x=-\frac{\sqrt3}{2}$

$2x=(-1)^{n}\cdot (-\frac{\pi}{3})+\pi n=(-1)^{n+1}\frac{\pi}{3}+\pi n,n\in Z\\\\b)\; cos2x+9sinx+4=0\\\\cos2x=cos^2x-sin^2x=(1-sin^2x)-sin^2x=1-2sin^2x\\\\1-2sin^2x+9sinx+4=0\\\\2sin^2x-9sinx-5=0\\\\D=81+40=121\\\\(sinx)_1=\frac{9-11}{4}=-\frac{1}{2}$

1\; net\; reshenij" alt="x=(-1)^{n}(-\frac{\pi}{6})+\pi n=(-1)^{n+1}\frac{\pi}{6}+\pi n,n\in Z\\\\(sinx)_2=\frac{9+11}{4}=5>1\; net\; reshenij" align="absmiddle" class="latex-formula">