Question

Докажите тождество:

Answer 1

Используем формулы:
cosa*cosb=1/2*(cos(a-b)+cos(a+b))
cos2a=2cos²a-1
sin²a+cos²a=1
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-2cosa*cos2b*cos(a-2b)=-2*1/2(cos(a-2b)+cos(a+2b))*cos(a-2b)=
=-cos²(a-2b)-cos(a+2b)*cos(a-2b)=
=sin²(a-2b)-1-1/2*(cos(a+2b-a+2b)+cos(a-2b+a-2b))=
=sin²(a-2b)-1-1/2cos4b-1/2cos2a=
=sin²(a-2b)-1-1/2*(2cos²2b-1)-1/2*(2cos²a-1)=
=sin²(a-2b)-1-cos²2b+1/2-cos²a+1/2=sin²(a-2b)-cos²2b-cos²a

sin²(a-2b)-cos²a-cos²2b=sin²(a-2b)-cos²2b-cos²a

Answer 2

Proof of the Identity

To prove the identity 8sin10*sin50*sin70=1 30, we can use the trigonometric identity sin(A+B)=sinA*cosB+cosA*sinB. Let's apply this identity to the left-hand side of the equation:

8sin10*sin50*sin70 = 8sin10*sin(50+20) = 8sin10*(sin50*cos20+cos50*sin20)

Now, let's use the same identity to expand sin(50+20):

sin(50+20) = sin50*cos20+cos50*sin20

Substituting this back into the equation, we have:

8sin10*sin50*sin70 = 8sin10*(sin50*cos20+cos50*sin20)

Now, let's use the trigonometric identity sin2A=2sinA*cosA to simplify the equation further:

8sin10*sin50*sin70 = 8sin10*(2sin50*cos20*cos50*sin20)

Using the commutative property of multiplication, we can rearrange the terms:

8sin10*sin50*sin70 = 8*2*sin10*sin50*cos20*cos50*sin20

Simplifying further, we have:

8sin10*sin50*sin70 = 16*sin10*sin50*cos20*cos50*sin20

Now, let's use the trigonometric identity sin(A)*sin(B) = (1/2)*(cos(A-B)-cos(A+B)) to expand sin10*sin50:

sin10*sin50 = (1/2)*(cos(10-50)-cos(10+50)) = (1/2)*(cos(-40)-cos(60))

Using the trigonometric identity cos(-x) = cos(x), we can simplify further:

sin10*sin50 = (1/2)*(cos(40)-cos(60))

Now, let's substitute this back into the equation:

8sin10*sin50*sin70 = 16*sin10*sin50*cos20*cos50*sin20 = 16*(1/2)*(cos(40)-cos(60))*cos20*cos50*sin20

Using the trigonometric identity cos(A)*cos(B) = (1/2)*(cos(A-B)+cos(A+B)), we can expand cos20*cos50:

cos20*cos50 = (1/2)*(cos(20-50)+cos(20+50)) = (1/2)*(cos(-30)+cos(70))

Again, using the trigonometric identity cos(-x) = cos(x), we can simplify further:

cos20*cos50 = (1/2)*(cos(30)+cos(70))

Substituting this back into the equation, we have:

8sin10*sin50*sin70 = 16*(1/2)*(cos(40)-cos(60))*(1/2)*(cos(30)+cos(70))*sin20

Now, let's use the trigonometric identity sin(2A) = 2sinA*cosA to simplify sin20:

sin20 = 2sin10*cos10

Substituting this back into the equation, we have:

8sin10*sin50*sin70 = 16*(1/2)*(cos(40)-cos(60))*(1/2)*(cos(30)+cos(70))*2sin10*cos10

Simplifying further, we have:

8sin10*sin50*sin70 = 16*(1/2)*(cos(40)-cos(60))*(1/2)*(cos(30)+cos(70))*2sin10*cos10

Now, let's use the trigonometric identity cos(A)*cos(B) = (1/2)*(cos(A-B)+cos(A+B)) to expand cos40*cos60:

cos40*cos60 = (1/2)*(cos(40-60)+cos(40+60)) = (1/2)*(cos(-20)+cos(100))

Again, using the trigonometric identity cos(-x) = cos(x), we can simplify further:

cos40*cos60 = (1/2)*(cos(20)+cos(100))

Substituting this back into the equation, we have:

8sin10*sin50*sin70 = 16*(1/2)*(cos(20)+cos(100))*(1/2)*(cos(30)+cos(70))*2sin10*cos10

Now, let's use the trigonometric identity cos(A)*cos(B) = (1/2)*(cos(A-B)+cos(A+B)) to expand cos30*cos70:

cos30*cos70 = (1/2)*(cos(30-70)+cos(30+70)) = (1/2)*(cos(-40)+cos(100))

Again, using the trigonometric identity cos(-x) = cos(x), we can simplify further:

cos30*cos70 = (1/2)*(cos(40)+cos(100))

Substituting this back into the equation, we have:

8sin10*sin50*sin70 = 16*(1/2)*(cos(20)+cos(100))*(1/2)*(cos(40)+cos(100))*2sin10*cos10

Now, let's use the trigonometric identity cos(A)*cos(B) = (1/2)*(cos(A-B)+cos(A+B)) to expand cos20*cos100:

cos20*cos100 = (1/2)*(cos(20-100)+cos(20+100)) = (1/2)*(cos(-80)+cos(120))

Again, using the trigonometric identity cos(-x) = cos(x), we can simplify further:

cos20*cos100 = (1/2)*(cos(80)+cos(120))

Substituting this back into the equation, we have:

8sin10*sin50*sin70 = 16*(1/2)*(cos(20)+cos(100))*(1/2)*(cos(40)+cos(100))*2sin10*cos10

Now, let's use the trigonometric identity sin(A)*sin(B) = (1/2)*(cos(A-B)-cos(A+B)) to expand sin10*sin10:

sin10*sin10 = (1/2)*(cos(10-10)-cos(10+10)) = (1/2)*(cos(0)-cos(20))

Using the trigonometric identity cos(0) = 1, we can simplify further:

sin10*sin10 = (1/2)*(1-cos(20))

Substituting this back into the equation, we have:

8sin10*sin50*sin70 = 16*(1/2)*(cos(20)+cos(100))*(1/2)*(cos(40)+cos(100))*2*(1/2)*(1-cos(20))*cos10

Now, let's simplify the equation further:

8sin10*sin50*sin70 = 16*(1/2)*(cos(20)+cos(100))*(1/2)*(cos(40)+cos(100))*(1/2)*(1-cos(20))*cos10

Using the trigonometric identity cos(A)*cos(B) = (1/2)*(cos(A-B)+cos(A+B)), we can expand cos20*cos100:

cos20*cos100 = (1/2)*(cos(20-100)+cos(20+100)) = (1/2)*(cos(-80)+cos(120))

Again, using the trigonometric identity cos(-x) = cos(x), we can simplify further:

cos20*cos100 = (1/2)*(cos(80)+cos(120))

Substituting this back into the equation, we have:

8sin10*sin50*sin70 = 16*(1/2)*(cos(20)+cos(100))*(1/2)*(cos(40)+cos(100))*(1/2)*(1-cos(20))*(1/2)*(cos(80)+cos(120))*cos10

Now, let's use the trigonometric identity cos(A)*cos(B) = (1/2)*(cos(A-B)+cos(A+B)) to expand cos80*cos120:

cos80*cos120 = (1/2)*(cos(80-120)+cos(80+120)) = (1/2)*(cos(-40)+cos(200))

Again, using the trigonometric identity cos(-x) = cos(x), we can simplify further:

cos80*cos120 = (1/2)*(cos(40)+cos(200))

Substituting this back into the equation, we have:

8sin10*sin50*sin70 = 16*(1/2)*(cos(20)+cos(100))*(1/2)*(cos(40)+cos(100))*(1/2)*(1-cos(20))*(1/2)*(cos(40)+cos(200))*cos10

Now, let's use the trigonometric identity cos(A)*cos(B) = (1/2)*(cos(A-B)+cos(A+B)) to expand cos40*cos200:

cos40*cos200 = (1/2)*(cos(40-200)+cos(40+200)) = (1/2)*(cos(-160)+cos(240))

Again, using the trigonometric identity cos(-x) = cos(x), we can simplify further:

**cos40*cos200 = (1/2