Question

Упростить выражения:

а) i$\frac{cos^4\alpha - sin^4\alpha }{cos^2\alpha -sin^2} - (cos\alpha -sin\alpha)^2$

б) $ctg\frac{21\pi }{4} * cos \frac{11\pi }{3}$

Answer 1

Ответ:

$\displaystyle \frac{cos^4a-sin^4a}{cos^2a-sin^2a}-(cosa-sina)^2=\\\\\\=\frac{(cos^2a-sin^2a)(\overbrace{cos^2a+sin^2a}^{1})}{cos^2a-sin^2a}-(\underbrace{cos^2a+sin^2a}_{1}-\underbrace{2sina\cdot cosa}_{sin2a})=\\\\\\=1-(1-sin2a)=sin2a$

$\displaystyle 2)\ \ ctg\frac{21\pi }{4}\cdot cos\frac{11\pi }{3}=ctg\Big(5\pi +\frac{\pi}{4}\Big)\cdot cos\Big(3\pi +\frac{2\pi}{3}\Big)=ctg\frac{\pi}{4}\cdot \Big(-cos\frac{2\pi}{3}\Big)=\\\\\\=-1\cdot \Big(-\frac{1}{2}\Big)=\frac{1}{2}$