Question

1) (1+ ctg B)^2 + (1 - ctg B )^2
2)tg x + cos x/1 + sin x
3)cos B/1 - sin b + 1-sin b/cos B
4) 1 + tg a/1+ ctg a

Answer 1

1) (1+ ctgβ)²+ (1 - ctgβ )²=1+2ctgβ+ctg²β+1-2ctgβ+ctg²β=
=2+2ctg²β=2(1+ctg²β)=1/sin²β;
2) $tgx+ frac{cosx}{1+sinx} = frac{tgx(1+sinx)+cosx}{1+sinx} = frac{tgx+ frac{sin ^{2}x }{cosx}+cosx }{1+sinx}= frac{sinx+sin ^{2}x+cos^{2}x}{cosx}* \ * frac{1}{1+sinx}= frac{1+sinx}{cosx} * frac{1}{1+sinx}= frac{1}{cosx};$
3) $frac{cos beta }{1-sin beta}+ frac{1-sin beta }{cos beta }= frac{cos^{2} beta +1-2sin beta +sin^{2} beta }{(1-sin beta )cos beta } = frac{2(1-sin beta )}{(1-sin beta )cos beta }= frac{2}{cos beta };$
4) $frac{1+tg alpha }{1+ctg alpha } = frac{cos alpha+sin alpha }{cos alpha } * frac{sin alpha }{sin alpha +cos alpha }= frac{sin alpha }{cos alpha } =tg alpha .$