Prove the following Cos2 /7+ cos 4 /7+ cos6 /7+ cos8 /7+ cos10 /7+ cos 12 /7= -1 - Maths - Trigonometric Functions

Question

Prove the following Cos2π/7+ cos 4π/7+ cos6π/7+
cos8π/7+ cos10π/7+ cos 12 π/7= -1

Anuradha Sharma · Accepted Answer

Hi, LHS, cos2π7+cos4π7+cos6π7+cos8π7+cos10π7+cos12π7=2sinπ7cos2π7+2sinπ7cos4π7+2sinπ7cos6π7+2sinπ7cos8π7+2sinπ7cos10π7+2sinπ7cos12π72sinπ7Now using 2sinAcosB=sin A-B+sinA-B=sinπ7+2π7+sinπ7-2π7+sinπ7+4π7+sinπ7-4π7+sinπ7+6π7+sinπ7-6π7+sinπ7+8π7+sinπ7-8π7+sinπ7+10π7+sinπ7-10π7+sinπ7+12π7+sinπ7-12π72sinπ7=sin3π7+sin-π7+sin5π7+sin-3π7+sin7π7+sin-5π7+sin9π7+sin-7π7+sin11π7+sin-9π7+sin13π7+sin-11π72sinπ7=sin3π7-sinπ7+sin5π7-sin3π7+sin7π7-sin5π7+sin9π7-sin7π7+sin11π7-sin9π7+sin13π7-sin11π72sinπ7=sin13π7-sinπ72sinπ7=2cos13π7+π72sin13π7-π722sinπ7   Uysing sinC-sinD = 2cosC+D2sinC-D2=2cosπsin6π72sinπ7 = - sin6π7sinπ7but it will not be -1 so kindly recheck the RHS

Prove the following Cos2π/7+ cos 4π/7+ cos6π/7+ cos8π/7+ cos10π/7+ cos 12 π/7= -1