Question

Cos 20 degree cos 40 degree cos 80 degree equals to 1 by 8 prove that

Answer 1

Step-by-step explanation:

cos 80°.cos 40°.cos 20°=1/8

=cos(A+B).cos (A-B)

=(cosA.cosB - sinA.sinB) . (cosA.cosB +sinA.sinB)

• (a+b).(a-b)=a^2-b^2

therefore this will become

= cos^2A.cos^2B-sin^2A.sin^2B

= cos^2B - sin^2A

• cos40°= cos(60°-20°)
• cos80°= cos(60°+20°)

= cos(60°+20°).cos(60°-20°). cos20°

= (cos^2 20°-sin^2 60°).cos 20°

• sin 60° = √3/2

= (cos^2 20° -3/4).cos20°

= cos^3 20° - 3cos20°/4

= (4 cos^3 20° -3cos20°)/4

FORMULA

• cos3A=4cos^3 A - 3cos A

= cos 3 . 20° /4

= cos60° / 4

• cos 60° = 1/2

= 1/2÷ 4

= 1/2× 1/4

= 1/8

L.H.S=R.H.S

Hence Proved