Question

if A is not an integral multiple often prove that cosA cos2A cos4A cos8A=sin16A/16sinA and hence deduce that cos 2π/15 cis4π/15 cos8π/15 cos16π/15=1/16​

Answer 1

Given : cosA cos2A cos4A cos8A=sin16A/16sinA

To Find : Prove

deduce that cos 2π/15 cis4π/15 cos8π/15 cos16π/15=1/16​

Solution

cosA cos2A cos4A cos8A=sin16A/16sinA

LHS =   cosA cos2A cos4A cos8A

multiply and divide by 2SinA

= 2SinAcosA cos2A cos4A cos8A/2SinA

2SinAcosA = Sin2A

= Sin2Acos2A cos4A cos8A/2SinA

Sin2Acos2A = Sin4A/2

= Sin4Acos4A cos8A/2SinA.2

= Sin4Acos4A cos8A/4SinA

Sin4Acos4A = Sin8A/2

= Sin8Acos8A/4SinA .2

=  Sin8Acos8A/8SinA

Sin8Acos8A = Sin16A/2

=  Sin16A/8SinA.2

= Sin16A/16SinA

= RHS

QED

Hence proved

cos 2π/15 cis4π/15 cos8π/15 cos16π/15  

 cosA cos2A cos4A cos8A=sin16A/16sinA

A = 2π/15

= sin16A/16sinA

= sin(32π/15)/16sin(2π/15)

= sin( 2π + 2π/15)/16sin(2π/15)

=  sin( 2π/15)/16sin(2π/15)

= 1/16

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