prove that tan^pi/16 + tan^2pi/16 + +tan^7pi/16 = 35 - Maths - Trigonometric Functions

Question

prove that tan^pi/16 + tan^2pi/16 + +tan^7pi/16 = 35

Priyanka Kedia · Accepted Answer

7π16=π2-π166π16=π2-2π165π16=π2-3π164π16=π4L
H S
=tan2π16+tan22π16+tan23π16+tan24π16+tan25π16+tan26π16+tan27π16
=tan2π16+tan22π16+tan23π16+tan2π4+tan2π2-3π16+tan2π2-2π16+tan2π2-π16
=tan2π16+tan22π16+tan23π16+1+cot23π16+cot22π16+cot2π16
=tan2π16+cot2π16+1+cot22π16+tan22π16+tan23π16+cot23π16
Solve first
bracket,tan2π16+cot2π16=tanπ16+cotπ162-2tanπ16cotπ16
=sinπ16cosπ16+cosπ16sinπ162-2
=1sinπ16cosπ162-2 =22sinπ16cosπ162-2
=2sinπ82-2 =81-cosπ4-2 =822-1-2 =822+1-2Similarly, 2nd
bracket=81-cosπ2-2=8-2=6And, 3rd
bracket=81-cos3π4=822-1-2Thus, equation 1 becomes,L H S
=822+1-2+6+1+822-1-2=35=R H S Hence Proved

prove that tan^pi/16 + tan^2pi/16 +.........+tan^7pi/16 = 35