Suppose `alpha,beta,gammaa n ddelta` are the interior angles of regular pentagon, hexagon, decagon, and dodecagon, respectively, then the value of `|cosalphasecbetacosgammacos e cdelta|` is _________

Answer

1

Explanation

Interior angle of regular polygon of side n is `(180^(@) - (360^(@))/(n))`
Hence, `alpha = 180^(@), beta = 120^(@), gamma = 144^(@), delta = 150^(@)`
`:. cos alpha = cos 108^(@) = - sin 180^(@) = - ((sqrt5 -1)/(4))`
`sec beta = sec 120^(@) = -2`
`cos gamma = cos 144^(@) = - cos 36^(@) = - ((sqrt5 + 1)/(4))`
`"cosec" delta = "cosec" 150^(@) = +2`
`:. |((sqrt5 -1)/(4)) (2) ((sqrt5 +1)/(4)) (-2)| = 1`