Find the absolute value of: sec (pi/7)+sec (3pi/7

Question

Find the absolute value of: sec (pi/7)+sec (3pi/7)+sec (5pi/7)

Vikas TU · Accepted Answer

Dear student

Dear student
cos(π/7) + cos(3π/7) + cos(5π/7)

Multuiply and divide by 2sin(π/7)

= 1/2sin(π/7) [ 2sin(π/7)cos(π/7) + 2sin(π/7)cos(3π/7) + 2sin(π/7)cos(5π/7) ]

Use the following

sin(2A) = 2sinAcosA

2sinAcosB = sin(A+B) + sin(A-B)

= 1/2sin(π/7) [ sin(2π/7) + sin(π/7+3π/7) + sin(π/7-3π/7) + sin(π/7+5π/7) + sin(π/7-5π/7) ]

= 1/2sin(π/7) [ sin(2π/7) + sin(4π/7) + sin(-2π/7) + sin(6π/7) + sin(-4π/7) ]

= 1/2sin(π/7) [ sin(2π/7) + sin(4π/7) - sin(2π/7) + sin(6π/7) - sin(4π/7) ]

= 1/2sin(π/7) [sin(6π/7)]

= 1/2sin(π/7) [sin(π - π/7)]

= [sin(π/7)]/2sin(π/7)

= 1/2

Find the absolute value of: sec (pi/7)+sec (3pi/7)+sec (5pi/7)