Question

1)вычислите cos2a*tg 4a,если sin a-cos a=1/корень из 2

2)упростить ctg a + (sin a/ 1+cos a)

3)вычислите cos^2a-cos^4a+sin^4a

Answer 1

1)$sin a-cos a=frac{1}{sqrt{2}}\ frac{1}{2}=(sin a-cos a)^2=sin^2 a - 2sin a cos a + cos^2 a = 1 - sin 2a\ sin 2a = frac{1}{2}\ cos 2a tan 4a = frac{cos 2a sin 4 a}{cos 4a} = frac{2 cos 2a sin 2 a cos 2a}{1- 2sin^2 2a}=\ =frac{2sin 2a (1-sin^2 2a)}{1-2sin^2 a} = frac{3}{2} = 1.5$

 

2)

$cot a + frac{sin a}{1+cos a}=frac{cos a}{sin a} + frac{sin a(1-cos a)}{1-cos^2 a}=\ =frac{cos a}{sin a}+frac{1-cos a}{sin a}=frac{1}{sin a}$

 

3)

$cos^2a-cos^4a+sin^4a = cos^2a- ((cos^2 a)^2 - (sin^2 a)^2)=\ =cos^2 a - (cos^2 a - sin^2 a)(cos^2 a + sin^2 a) = sin^2 a$