Question and Answers Forum

All Questions      Topic List

Others Questions

Previous in All Question      Next in All Question      

Previous in Others      Next in Others      

Question Number 189375 by moh777 last updated on 15/Mar/23

      show that :      (1/(cscx + cot x)) = cscx − cot x

$$ \\ $$$$\:\:\:\:{show}\:{that}\:: \\ $$$$\:\:\:\:\frac{\mathrm{1}}{{cscx}\:+\:{cot}\:{x}}\:=\:{cscx}\:−\:{cot}\:{x} \\ $$$$ \\ $$$$ \\ $$

Answered by Ar Brandon last updated on 15/Mar/23

(1/(cscx+cotx))=((sinx)/(1+cosx))=((sinx(1−cosx))/(1−cos^2 x))  =((sinx(1−cosx))/(sin^2 x))=((1−cosx)/(sinx))  =cscx−cotx

$$\frac{\mathrm{1}}{\mathrm{csc}{x}+\mathrm{cot}{x}}=\frac{\mathrm{sin}{x}}{\mathrm{1}+\mathrm{cos}{x}}=\frac{\mathrm{sin}{x}\left(\mathrm{1}−\mathrm{cos}{x}\right)}{\mathrm{1}−\mathrm{cos}^{\mathrm{2}} {x}} \\ $$$$=\frac{\mathrm{sin}{x}\left(\mathrm{1}−\mathrm{cos}{x}\right)}{\mathrm{sin}^{\mathrm{2}} {x}}=\frac{\mathrm{1}−\mathrm{cos}{x}}{\mathrm{sin}{x}} \\ $$$$=\mathrm{csc}{x}−\mathrm{cot}{x} \\ $$

Answered by BaliramKumar last updated on 15/Mar/23

LHS = (1/(cosecx + cotx)) = ((cosec^2 x − cot^2 x)/(cosecx + cotx))               = (((cosecx + cotx)(cosecx − cotx))/((cosecx + cotx)))                = cosecx − cotx = RHS

$${LHS}\:=\:\frac{\mathrm{1}}{{cosecx}\:+\:{cotx}}\:=\:\frac{{cosec}^{\mathrm{2}} {x}\:−\:{cot}^{\mathrm{2}} {x}}{{cosecx}\:+\:{cotx}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:=\:\frac{\cancel{\left({cosecx}\:+\:{cotx}\right)}\left({cosecx}\:−\:{cotx}\right)}{\cancel{\left({cosecx}\:+\:{cotx}\right)}}\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:=\:{cosecx}\:−\:{cotx}\:=\:{RHS} \\ $$$$ \\ $$

Answered by manxsol last updated on 15/Mar/23

recall     cosec^2 =1+cot^2 x   (1/(cscx + cot x)) ×((cscx−cotx)/(cscx−cotx))=  ((cscx−cotx)/(csc^2 x−ct^2 x))=cscx−cotx

$${recall}\:\:\:\:\:{cosec}^{\mathrm{2}} =\mathrm{1}+{cot}^{\mathrm{2}} {x} \\ $$$$\:\frac{\mathrm{1}}{{cscx}\:+\:{cot}\:{x}}\:×\frac{{cscx}−{cotx}}{{cscx}−{cotx}}= \\ $$$$\frac{{cscx}−{cotx}}{{csc}^{\mathrm{2}} {x}−{ct}^{\mathrm{2}} {x}}={cscx}−{cotx} \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com