Question

докажите тождество ((1-4sin^2α*cos^2α)/(sinα+cosα)^2) +2sinα*cosα=1

Answer 1

$\dfrac{1-4\sin^2\alpha\cos^2\alpha}{(\sin\alpha+\cos \alpha)^2}+2\sin\alpha\cos \alpha=\dfrac{1-\sin^22\alpha}{1+\sin 2\alpha}+\sin 2\alpha=\\ \\ =\dfrac{(1-\sin2\alpha)(1+\sin2\alpha)}{1+\sin2\alpha}+\sin2\alpha=1-\sin2\alpha+\sin2\alpha=1$