Question

三角数列题：sinθ sinα cosθ成等差数列,sinθ sinβ cosθ为等比数列,求证2COS2α=cos2β

Answer 1

证明：∵sinθ sinα cosθ成等差数列∴2sina=sinθ+cosθ,

∵sinθ sinβ cosθ为等比数列,∴sin^2β=sincθosθ,

∴左边=2cos2a=2[1-2(sina)^2]=2-4(sina)^2=2-(sinθ+cosθ)^2=2-(1+2sinθcosθ)=1-2sinθcosθ

=1-2(sinβ)^2=cos2β=右边