806. а) sin(90°−α)+cos(180°+α)+tg(270°+α)+ctg(360°+α)= =cosα+(−cosα)+(−ctgα)+ctgα=0.
⎛π ⎞ ⎛ 5π ⎞ + α ⎟⎟ − cos(α − π ) + tg (π − α ) + ctg⎜⎜ − α ⎟⎟ = ⎝2 ⎠ ⎝ 2 ⎠
б) sin ⎜⎜
⎛π ⎞ = cosα − cos(π − α ) − tgα +ctg ⎜ − α ⎟ = 2 ⎝ ⎠ cos α − ( − cos α ) − tgα + tgα = 2 cos α . 807. а)
cos( −α ) cos(180° + α ) cos α ⋅ (− cos α ) = ctg α. = sin( −α ) sin( 90° + α ) − sin( −α ) ⋅ cos α
sin(π + α ) cos(2π − α ) − sin α ⋅ cos α = = tg(π − α ) cos(α − π ) − tgα cos(− cos α ) sin α ⋅ cos α ⋅ cos α =− = − cos α . sin α ⋅ cos α sin(−α )ctg(−α ) sin α ⋅ ctgα = = в) cos(360° − α )tg(180° + α ) cos α ⋅ tgα sin α ⋅ cos α ⋅ cos α cos α = = = ctgα . cos α ⋅ sin α ⋅ sin α sin α sin(π + α )sin(α + 2π ) − sin α ⋅ sin α г) = = tg (π + α )cos (1,5π + α ) tgα ⋅ sin α б)
=−
sin α ⋅ cos α = − cosα . sin α
808. а) sin2(180°−x)+sin2(270°−x)=sin2x+(–cosx)2=sin2x+cos2x=1.
⎛
б) sin(π−x) cos ⎜ x −
⎝
π⎞ π⎞ ⎛ ⎟ − sin ⎜ x + ⎟ cos(π−x)= 2⎠ 2⎠ ⎝
=sin x⋅sin x − cos x ⋅ (− cos x) = sin2x+cos2x=1. 809. Воспользуемся формулами приведения.
⎞ ⎛π + x ⎟ = (− cos x) 2 + (− sin x) 2 = ⎠ ⎝2
а) cos2(π+x)+cos2 ⎜
= cos 2 x + sin 2 x = 1. ⎛π ⎞ ⎛ 3π ⎞ б) sin(π+x)cos ⎜ + x ⎟ − cos(2π + x ) sin ⎜ −x⎟ = ⎠ ⎝ 2 ⎠ ⎝2 =−sin x⋅(−sin x)−cos x⋅(−cos x)=sin2x+cos2x=1. 267