Question

PLEASE HELP Verify the identity. tan x + pi/2 = -cot x

Answer 1

\[\tan (x+\dfrac {\pi} {2})=-\cot (x)\] using the identity for cos(x+y)=cosxcosy-senxseny and the identity sen(x+y)=senxcosy+cosxseny, we will have
\[\tan(x+\dfrac {\pi} {2})=\dfrac {sen(x+\dfrac {\pi} {2})} {\cos(x+\dfrac {\pi} {2})}=\dfrac {sen(x)\cos(\dfrac {\pi} {2})+\cos(x)sen((\dfrac {\pi} {2}))} {\cos(x)\cos(\dfrac {\pi} {2})-sen(x)sen(\dfrac {\pi} {2})}\] cos(pi/2)=0 and sen (pi/2)=1
\[\dfrac {sen(x)\cos(\dfrac {\pi} {2})+\cos(x)sen((\dfrac {\pi} {2}))} {\cos(x)\cos(\dfrac {\pi} {2})-sen(x)sen(\dfrac {\pi} {2})}=\dfrac {\cos(x)} {-sen(x)}=-\cot(x)\]