Question

Докажите тождество :
(1-cos2a)/sin2a=tga
Упростите:
1+(cosa tg^2 a)/1+cosa

Answer 1

(1-cos2α)/sin2α = (1-cos2α)/(2sinαcosα) = (1-cos2α)/2 · 1/(sinαcosα) =

= sin²α· 1/(sinαcosα) = sin²α/(sinαcosα) = sinα/cosα = tgα.

1+(cosαtg²α)/(1+cosα) = 1 + (cosα(1/cos²α - 1))/(1+cosα) = 1 + (cosα(1-cos²α)/cos²α)/(1+cosα) = 1 + (cosα(1-cosα)(1 + cosα))/(cos²α(1+cosα)) =

=1 + (1-cosα)/(cosα) = (cosα + 1 - cosα)/cosα = 1/cosα