Question

1.$frac{tgx+tgy}{tg(x+y)}+ frac{tgx-tgy}{tg(x-y)}=$
2.$sin(pi-2 alpha )+2cos( frac{3}{2}pi+ alpha )sin( frac{3}{2}pi- alpha )=$
3.$cos^{4} frac{ alpha }{2}-sin^{4} frac{ alpha }{2}$

Answer 1

1. tg (a + b) = (tg a + tg b) / (1 - tg a tg b)
tg (a - b) = (tg a - tg b) / (1 + tg a tg b)
(tg x + tg y)/tg(x + y) + (tg x - tg y) / (tg(x-y) = (tg x + tg y) / (tg x + tg y) * (1 - tg x tg y) + +(tg x - tg y) / (tg x - tg y) * (1 + tg x tg y) = 1 - tg x tg y + 1 + tg x tg y = 2

2. sin (π - 2a) + 2 cos(3/2π + a)sin(3/2π-a) = sin 2a + 2 sin a (-cos a) = 2 sin a * cos a - 2 sin a cos a = 0

3. cos^4 a/2 - sin^4 a/2 = (cos ² a - sin ²a)(cos² a + sin² a) = cos²a - sin²a = cos 2a

Answer 2

1
(tgx+tgy((1-tgxtgy)/(tgx+tgy)+(tgx-tgy)(1+tgxtgy)/(tgx-tgy)=1-tgxtgy+1+tgxtgy=2
2
sin2a-2sinacosa=sin2a-sin2a=0
3
(cos²a/2-sin²a/2)(cos²a/2+sin²a/2=cosa*1=cosa