Question

设sina是sinΘ,cosΘ的等差中项sinβ是sinΘ,cosΘ的等比中项.求证cos4-4cos4a=3

Answer 1

由已知2sina=sinΘ+cosΘ (1)

(sinβ)^2=sinΘcosΘ (2)

(1)^2 4(sina)^2=1+2sinΘcosΘ=1+2(sinβ)^2

2cos2α=cos2β

平方得4(cos2α)^2=(cos2β)^2

2(cos4α+1)=(1/2)(cos4β+1)

故cos4β-4cos4a=3