Question

demostrar las siguientes entidades por favor

1) cos α tan α = sen α
2) sen α sec α = tan α
3)sen α cot α = cos α
4) sen α tan α + cos α = sec α
5) cosec α - sen α =cot αcos α

Answer 1

1)
${cos(x).tan(x)=}\\\\{\frac{cos(x)}{1}.\frac{sen(x)}{cos(x)}=}\\\\{\boxed{sen(x)}}$

2.)
${sen(x).sec(x)=}\\\\{\frac{sen(x)}{1}.\frac{1}{cos(x)}=}\\\\{\frac{sen(x)}{cos(x)}=} \\\\{\boxed{tan(x)}}$

3.)
${sen(x).ctg(x)=}\\\\{\frac{sen(x)}{1}.\frac{cos(x)}{sen(x)}=}\\\\{\boxed{cos(x)}}$

4.)
${sen(x).tan(x)+cos(x)=}\\\\{\frac{sen(x)}{1}.\frac{sen(x)}{cos(x)}+cosx=}\\\\{\frac{sen^2(x)}{cosx}+cos(x)=}\\\\{\frac{sen^2(x)+cos^2(x)}{cos(x)}=}\\\\{\frac{1}{cos(x)}=}\\\\{\boxed{sec(x)}}$

5.)
${csc(x)-senx(x)=}\\\\{\frac{1}{senx}-senx(x)=}\\\\{\frac{1-sen^2(x)}{sen(x)}=}\\\\{\frac{cos^2(x)}{sen(x)}=}\\\\{\frac{cos(x)}{sen(x)}.\frac{cos(x)}{1}=}\\\\{\boxed{ctg(x).cos(x)}}\\\\{\mathbf{salu2.!!\ :)}}\\{\mathbf{Wellington}}$