Question

Verify the identity


`cot(x-(\pi )/(2) )` = - tan x

please show all steps

Answer 1

Explanation:

`\tan (x-(\pi )/(2) )=(tan~x-tan~(\pi )/(2) )/(1+tan~x~tan~(\pi )/(2) ) \\or~\cot(x-(\pi )/(2) )=(1+\tan~x~\tan(\pi )/(2) )/(\tan~x-\tan~(\pi )/(2) ) \\divide~the~R.H.S. ~by~\tan ~x~\tan~(\pi )/(2) \\=(\cot~x~\cot~(\pi )/(2) +1 )/(\cot~(\pi )/(2) -\cot~x) \\\cot(\pi )/(2) =0\\\cot(x-(\pi )/(2) )=(\cot~x ~*~0+1)/(0-\cot~x) \\\cot~(x-(\pi )/(2) )=-(1)/(\cot~x) =-\tan~x`