Verified answer

1)3cos2x+sin^2x+5sinx cosx=0  (cos2x=cos^2x -sin^2x)

3cos^2x -3 sin^2x+sin^2x+5sinx cosx=0  (/cos^2x )

3+5tgx-2tgx^2x=0

2tgx^2x-5tgx-3=0

Пусть tgx=t

2t^2-5t-3=0

D=49

t1=3      

tgx=3        

x=arctg3+ πn, n∈Z

t2=1/2    

 tgx=1/2    

x=arctg1/2+ πn, n∈Z  

2)sin 7x - sin x= cos 4 x       (sinα-sinβ=2sin(α-β)/2cos(α+β)/2)

2sin3xcos4x-cos4x=0

cos4x(2sin3x-1)=0

1. cos 4x=0        

 4x=π/2+πn, n∈Z      

x=π/8+πn/4, n∈Z

2. 2sin3x-1=0    

sin3x=1/2      

3x=(-1)^k arsin1/2+ πk, k∈Z     

x=(-1)^k π/18+ πk/3, k∈Z

3)cosx + cos 3 x=4cos2x      (cosα+cosβ=2cos(α+β)/2cos(α-β)/2)

2cos2xcosx=4cos2x

cos2xcosx=2cos2x

cos2x(cosx-2)=0

1. cos2x=0    

2x=π/2+πn, n∈Z       

x=π/4+πn/2, n∈Z

2. cosx≠2

4)1-sin x cos x +2 cos^2x=0                       (1=cos^2x +sin^2x)

cos^2x +sin^2x-sin x cos x +2 cos^2x=0    (/cos^2x )

1 + tg^2x - tgx +2 = 0

 tg^2x - tgx+3=0

D=1-12≠ -11

5)arcsin 1 /√2 - 4arcsin 1=0

(arcsin1=π/2, cos(arcsin1)=cosπ/2=0).

6)arccos(-1) - arcsin (-1)=

=π-arccosx - arcsinx = 

=π-(arccosx + arcsinx)=              (arccosx + arcsinx=π/2)

=π- π/2=  π/2

7)4arctg(-1)+3 arctg√ 3 =      

=4(-π/4)+3(π/6)=                    (tg45=1 (45 °=π/4 ).  tg(-45)=-1. arctg(-1)=-45=-π/4

 =-π+π/2= -π/2                        tg30=√3   arctg√3=30°=π/6)