1. докажите тождество. sin (a+b) + sin (a-b) = 2sina cosb 2. известно, чт cos (п/4 + t) + cos (п/4 - t) = p. найдите cos (п/4 + t) cos (п/4-t

1. sin(a+b)+sin(a-b)=2sina cosb

sina*cosb+cosa*sinb+sina*cosb-cosa*sinb=2sina cosb

 

Cosa - cos10a - (cos4a - cos7a) = -2sin(5.5a)sin(-4.5a) + 2sin(5.5a)sin(-1.5a) = 2sin(5.5a) *  (sin4.5a - sin1.5a); 1) sina + sin10a + sin4a + sin7a = 2sin5.5a * cos4.5a + 2sin5.5a * cos1.5a = 2sin5.5a * (cos4.5a   + cos1.5a);   2) 1) / 2) = [2sin(5.5a) *  (sin4.5a - sin1.5a)] / [2sin5.5a * (cos4.5a   + cos1.5a)] =  (sin4.5a - sin1.5a)  /  (cos4.5a   + cos1.5a ); sin4.5a - sin1.5a = 2cos3a * sin1.5a; 3)cos4.5a + cos1.5a = 2cos3a * cos1.5a; 4)3) / 4) = [2cos3a * sin1.5a]  / [ 2cos3a * cos1.5a ] = tg1.5a

Решение 1) y(t)=(t^2-1)/(t^2+1)  y`(t) = [2t*(t² + 1) - 2t*(t² - 1)] / (t² + 1)² = (2t³ + 2t - 2t³ + 2t) /    (t² + 1)² == 4t /    (t² + 1)² 2) y(t) =  (t^3-3t+1)/(t^3+2)  y`(t) = [(3t² - 3)*(t³ + 2) - 3t² * (t³ - 3t + 1)] / (t³ + 2)²  =  (3t⁵ + 6t²- 3t³ - 6 - 3t⁵ + 9t³ + 3t²) /  (t³ + 2)² =  (6t³ + 9t² - 6)  /  (t³ + 2)² 3)  f(x)  =  (2+1/x^2)/(3-5x^3) f`(x) = [(- 2/x³)*(3 - 5x³) - 15x² * (2 + 1/x²)] / (3 - 5x)²    4) y(t)  =  √t (3^√t +2t)  y`(t) = (1/2√t)*  (3^√t +2t)   +  √t [ 3^√t * ln3 * (1/2 √t)   + 2]