Question

Докажите тождество. (! решите все 3, )

Answer 1

$1)\; \; \frac{1+2\, sint\, cost}{(sint+cost)^2}=\frac{1+2\, sint\, cost}{sin^2t+cos^2t+2\, sint\, cost}=\frac{1+2\, sint\, cost}{1+2\, sint\, cost}=1\\\\\\2)\; \; sint+\frac{cos^2t}{1+sint} =\frac{sint\, (1+sint)+cos^2t}{1+sint}=\frac{sint+sin^2t+cos^2t}{1+sint}=\frac{sint+1}{1+sint}=1\\\\\\3)\; \; \frac{(2\, sin3t-2\, cos3t)^2}{2-4\, sin3t\, cos3t}=\frac{2^2\cdot (sin3t-cos3t)^2}{2\cdot (1-2sin3t\, cos3t)}=\frac{2\cdot (sin^23t+cos^23t-2\, sin3t\, cos3t)}{1-2sin3t\, cos3t}=\\\\=\frac{2\cdot (1-2\, sin3t\, cos3t)}{1-2\, sin3t\, cos3t}=2$