Question

Вычислите; tg3-tg 48)/(1+tg 3*ctg 42)

Answer 1

$frac{tan 3-tan 48}{1+tan 3*cot 42}=frac{frac{sin 3}{cos 3}-frac{sin 48}{cos 48}}{1+frac{sin 3}{cos 3}*frac{cos 42}{sin 42}}=\ =frac{sin 3 cos 48-sin 48cos 3}{cos 3 sin 42+sin 3cos 42}*frac{cos 3 sin 42}{cos 3 cos 48}=\ frac{sin(48-3)}{sin(3+42)}*frac{sin 42}{cos 48}=frac{sin 45}{sin 45} *frac{sin 42}{cos 48}= frac{sin (45-3)}{cos (45+3)}=\ = frac{sin 45cos3-cos 45 sin 3}{cos 45cos3-sin 45 sin 3}=$

 

$= frac{frac{sqrt{2}}{2}(cos 3 - sin 3)}{frac{sqrt{2}}{2}(cos 3 - sin 3)} = 1$