How do you verify tan (x-(pi/2)) = cot x? Thanks.

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Answer 1 · 2018-04-07T22:17:16

See below. #tan(x-pi/2)=-cotx#, not #cotx.#

We cannot use the tangent difference identity, as #tan(pi/2)# doesn't exist.

Instead, you can recall that #tantheta=sintheta/costheta # and therefore

#tan(x-pi/2)=sin(x-pi/2)/cos(x-pi/2)# and use the sine and cosine differences formulas:

#sin(x-y)=sinxcosy-cosxsiny#

#sin(x-pi/2)=sinxcos(pi/2)-cosxsin(pi/2)#

#cos(pi/2)=0, sin(pi/2)=1#

#sin(x-pi/2)=-cosx#

#cos(x-y)=cosxcosy+sinxsiny#

#cos(x-pi/2)=cosxcos(pi/2)+sinxsin(pi/2)#

#cos(x-pi/2)=sinx#

Thus,

#tan(x-pi/2)=-cosx/sinx=-cotx#