Question

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Answer 1

$frac{1+2sin beta cos beta }{(sin beta +cos beta )^2} = frac{1+2sin beta cos beta }{sin^2 beta+2sin beta cos beta +cos^2 beta } = frac{1+2sin beta cos beta }{1+2sin beta cos beta }=1$

$frac{sin^2 beta - cos^2 beta +1 }{sin^2 beta } = frac{sin^2 beta + sin^2 beta }{sin^2 beta } = frac{2sin^2 beta }{sin^2 beta } = 2$

$frac{1}{1+tg^2 beta } + frac{1}{1+ctg^2 beta }= frac{1}{1+ frac{sin^2 beta }{cos^2 beta } } + frac{1}{1+ frac{cos^2 beta }{sin^2 beta } }= frac{1}{frac{cos^ beta +sin^2 beta }{cos^2 beta } } + frac{1}{ frac{sin^2 beta +cos^2 beta }{sin^2 beta } }= \ = frac{1}{frac{1 }{cos^2 beta } } + frac{1}{ frac{1}{sin^2 beta } }=cos^2 beta +sin^2 beta =1$

$frac{1+sin beta }{cos beta} * frac{1-sin beta }{cos beta} = frac{(1+sin beta)(1-sin beta )}{cos^2 beta } = frac{1-sin^2 beta }{cos^2 beta } = frac{cos^2 beta }{cos^2 beta } =1$

(sinα+cosα)²-2sinαcosα=sin²α+2sinαcosα+cos²α-2sinαcosα=sin²α+cos²α=1

sin⁴α+cos⁴α+2sin²αcos²α=(sin²α+cos²α)²=1²=1

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

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