Question

Докажите тождество
1) 1+2sinacosa / (sina+cosa)^2 =1
2) sin^2a-cos^2a+1 / sin^2a=2
3) (2-sina)(2+sina)+(2-cosa)(2+cosa)=7

Answer 1

$3) (2-sin alpha)(2+sin alpha)+ (2-cos alpha)(2+cos alpha)=7 \ 4-sin^{2} alpha +4-cos^{2} alpha =7 \ 8-(sin^{2} alpha +cos^{2} alpha )=7 \ 8-1=7 \ 7=7$$1) frac{1+2sin alpha cos alpha }{(sin alpha +cos alpha )^{2}}=1 \ frac{1+2sin alpha cos alpha }{(sin^{2} alpha +2sin alpha cos alpha + cos^{2} alpha )}=1 \ frac{1+2sin alpha cos alpha }{(1 +2sin alpha cos alpha)}=1 \ 1=1 \ 2) frac{sin^{2} alpha -cos^{2} alpha +1}{sin^{2} alpha } =2 \ frac{sin^{2} alpha -cos^{2} alpha +sin^{2} alpha +cos^{2} alpha}{sin^{2} alpha } =2 \ frac{2sin^{2} alpha }{sin^{2} alpha } =2 \ 2=2$

Answer 2

1) 1+2sinacosa / (sina+cosa)^2 =(sin²a+cos²a+2sinacosa)/(sin²a+2sinacosa+cos²a)=1
2) sin^2a-cos^2a+1 / sin^2a=(sin²a-cos²a+sin²a+cos²a)/sin²a=2sin²a/sin²a=2
3) (2-sina)(2+sina)+(2-cosa)(2+cosa)=4-sin²a+4-cos²a=8-(sin²a+cos²a)=8-1=7