Question

Решите пожалуйста четвёртый и пятый номера.

Answer 1

4) $ctg^{2} \alpha ctg^{2} \beta -\frac{cos^{2}(\alpha -\beta )+cos^{2}(\alpha +\beta )}{2sin^{2}(-\alpha) sin^{2}\beta} = ctg^{2} \alpha ctg^{2} \beta -\frac{cos^{2}\alpha cos^{2}\beta+sin^{2}\alpha sin^{2}\beta +cos^{2}\alpha cos^{2}\beta-sin^{2}\alpha sin^{2}\beta}{2sin^{2}\alpha sin^{2}\beta} =ctg^{2} \alpha ctg^{2} \beta -\frac{2cos^{2}\alpha cos^{2}\beta}{2sin^{2}\alpha sin^{2}\beta} = ctg^{2} \alpha ctg^{2} \beta-ctg^{2} \alpha ctg^{2} \beta = 0$

5) sinα - cosα = √2/2; (sinα - cosα)² = (√2/2)²; cos²α - 2cosαsinα + sin²α = 1/2;

1 - 2cosαsinα = 1/2; cosαsinα = 1/4

cos³α - sin³α = ( cosα - sinα)( cos²α + cosαsinα + sin²α) = ( cosα - sinα)( 1 + cosαsinα) = -√2/2(1 + 1/4) = -√2/2 · 5/4 = (-5√2)/8.