How do you express cos(4π3)cos(7π4) without using products of trigonometric functions?

1 Answer
Oct 4, 2016

cos(4π3)cos(7π4)=12cos(13π12)+12cos(5π12)

Explanation:

As cos(A+B)=cosAcosBsinAsinB and

cos(AB)=cosAcosB+sinAsinB

cos(A+B)+cos(AB)=2cosAcosB

or cosAcosB=12[cos(A+B)+cos(AB)]

Hence cos(4π3)cos(7π4)

= 12[cos((4π3)+(7π4))+cos((4π3)(7π4))]

= 12[cos(4×4π+3×7π12)+cos(4×4π3×7π12)]

= 12[cos(16π+21π12)+cos(16π21π12)]

= 12[cos(37π12)+cos(5π12)]

= 12[cos(2π+13π12)+cos(5π12)]

= 12cos(13π12)+12cos(5π12)