Question

Доказать тождество:
2sin2a-sin4a/2sin2a+sin4a=tg^2

Answer 1

2sin2a-sin4a/2sin2a+sin4a=2sin2a-2sin2acos2x/2sin2a+2sin2acos2a=2sin2a(1-cos2a)/3sin2a(1+cos2a)=(1-cos2a)/(1+cos2a)=(1-(1-2sin²a))/(1+(2cos²a-1))=2sin²a/2cos²a=tg²a
sin 2x=2 sinx cosx
cos2x=2cos²-1=1-2sin²x

Answer 2

$frac{2sin2a-sin4a}{2sin2a+sin4a} = frac{2sin2a-2sin2acos2a}{2sin2a+2sin2acos2a} = \ frac{2sin2a(1-cos2a)}{2sin2a(1+cos2a} = frac{1-cos2a}{1+cos2a}= \ frac{sin^{2}a+cos^{2}a-cos^{2}a+sin^{2}a}{sin^{2}a+cos^{2}a+cos^{2}a-sin^{2}a} = \ frac{2sin^{2}a}{2cos^{2}a} =tg^{2}a$