sec−1⎛⎜
⎜⎝1410∑k=01cos(7π12+kπ2)cos(7π12+(k+1)π2)⎞⎟
⎟⎠
=sec−1⎛⎜
⎜⎝1410∑k=0sin[(7π12+(k+1)π2)− (7π12+kπ2)]cos(7π12+kπ2)cos(7π12+(k+1)π2)⎞⎟
⎟⎠
(∵sin(A−B)=sinA⋅cosB−cosA⋅sinB)
=sec−1⎛⎜
⎜⎝1410∑k=0(sin(7π12+(k+1)π2)cos(7π12+kπ2))−(cos(7π12+(k+1)π2)sin(7π12+kπ2))cos(7π12+kπ2)cos(7π12+(k+1)π2)⎞⎟
⎟⎠
=sec−1(1410∑k=0(tan(7π12+(k+1)π2))−(tan(7π12+kπ2)))
=sec−114(tan(7π12+π2)−tan(7π12)+tan(7π12+2π2)−tan(7π12+π2)+ ....+tan(7π12+11π2)−tan(7π12+10π2))
=sec−114(tan(7π12+11π2)−tan7π12)
=sec−114(tan(6π+π12)−tan(π−5π12))
=sec−114(tanπ12+tan5π12)
=sec−114[(2−√3)+(2+√3)]
=sec−1(1)=0