Решить уравнения:

cos2x-5cosx-2=0

1-cos8x=sin 4x

sin^2x+4sinx*cosx+3cos^2x=0

cos4x-sin4x=-1/2

xsin^2x+sin^2 3x+sin^2 4x+sin^2 5x=2

1) cos2x-5cosx-2=0; ⇒

2cos²x-1-5cosx-2=0; ⇒cosx=y; -1≤y≤1; ⇒

2y²-5y-3=0;

y₁,₂ = (5⁺₋√ (25+24)) / 4 = (5⁺₋7) / 4;

y₁ = (5+7) / 4=3; ⇒y₁>1⇒нет решения;

y₂ = (5-7) / 4=-1/2; ⇒

cosx=-1/2; ⇒x=⁺₋2π/3+2kπ; k∈Z.

2) 1-cos8x=sin4x; ⇒

sin²4x+cos²4x-cos²4x+sin²4x=sin4x; ⇒

2sin²4x-sin4x=0; ⇒

sin4x (2sin4x-1) = 0; ⇒

sin4x=0⇒4x=nπ; k∈Z; ⇒x=nπ/4; n∈Z;

2sin4x-1=0; ⇒

sin4x=1/2;

4x = (-1) ⁿ·π/6+nπ; n∈Z.

3) sin²x+4sinxcosx+3cos²x=0; ⇒cos²x≠0 делим на cos²x:

tg²x+4tgx+3=0; tgx=y; ⇒

y²+4y+3=0;

y₁,₂=-2⁺₋√ (4-3) = - 2⁺₋1;

y₁=-1; ⇒

tgx=-1; ⇒x=-π/4+nπ; n∈Z;

y₂=-3; ⇒x=arctg (-3) + nπ; n∈Z.

4) cos4x-sin4x=-1/2; ⇒cos4x=sin4x-1/2; ⇒

cos²4x=sin²4x-2/2·sin4x+1/4; ⇒

1-sin²4x-sin²4x+sin4x-1/4=0⇒

-2sin²4x+sin4x+3/4=0; ⇒

sin4x=y; -1≤y≤1;

2y²-y-3/4=0;

y₁,₂ = (1⁺₋√ (1+6)) / 4 = (1⁺₋√7) / 4;

y₁ = (1+√7) / 4 = (1+2.646) / 4=0.9115;

sin4x=0.9115; ⇒4x = (-1) ⁿarcsin (0.9115) + 2nπ; n∈Z;

x = (-1) ⁿ (arcsin (0.9115)) / 4+nπ/2; n∈Z;

y₂ = (1-2.646) / 4=-0.4115;

4x = (-1) ⁿarcsin (-0.4115) + 2nπ; n∈Z

x=[ (-1) ⁿarcsin (-0.4115) + 2nπ]/4.