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tg-2-590-cos-2-320-sin-111-cos-159-cos-279-sin-549-ctg-950-sin-2-400-simplify-

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Question Number 59811 by ANTARES VY last updated on 15/May/19
(((tg^2 (590°))/(cos^2 (320°)))+((sin(111°))/(cos(159°))))(((cos(279°))/(sin(549°)))+((ctg(950°))/(sin^2 (400°)))) simplify.
$$\left(\frac{\boldsymbol{\mathrm{tg}}^{\mathrm{2}} \left(\mathrm{590}°\right)}{\boldsymbol{\mathrm{cos}}^{\mathrm{2}} \left(\mathrm{320}°\right)}+\frac{\boldsymbol{\mathrm{sin}}\left(\mathrm{111}°\right)}{\boldsymbol{\mathrm{cos}}\left(\mathrm{159}°\right)}\right)\left(\frac{\boldsymbol{\mathrm{cos}}\left(\mathrm{279}°\right)}{\boldsymbol{\mathrm{sin}}\left(\mathrm{549}°\right)}+\frac{\boldsymbol{\mathrm{ctg}}\left(\mathrm{950}°\right)}{\boldsymbol{\mathrm{sin}}^{\mathrm{2}} \left(\mathrm{400}°\right)}\right) \\ $$$$\boldsymbol{\mathrm{simplify}}. \\ $$
Commented by Kunal12588 last updated on 15/May/19
sir is it (((tg^2 (590°))/(cos^2 (320°)))+((sin(111°))/(cos(159°))))(((cos(279°))/(sin(549°)))+((ctg^2 (950°))/(sin^2 (400°))))?
$${sir}\:{is}\:{it} \\ $$$$\left(\frac{\boldsymbol{\mathrm{tg}}^{\mathrm{2}} \left(\mathrm{590}°\right)}{\boldsymbol{\mathrm{cos}}^{\mathrm{2}} \left(\mathrm{320}°\right)}+\frac{\boldsymbol{\mathrm{sin}}\left(\mathrm{111}°\right)}{\boldsymbol{\mathrm{cos}}\left(\mathrm{159}°\right)}\right)\left(\frac{\boldsymbol{\mathrm{cos}}\left(\mathrm{279}°\right)}{\boldsymbol{\mathrm{sin}}\left(\mathrm{549}°\right)}+\frac{\boldsymbol{\mathrm{ctg}}^{\mathrm{2}} \left(\mathrm{950}°\right)}{\boldsymbol{\mathrm{sin}}^{\mathrm{2}} \left(\mathrm{400}°\right)}\right)? \\ $$
Commented by ANTARES VY last updated on 15/May/19
yes
$$\boldsymbol{\mathrm{yes}} \\ $$
Answered by Kunal12588 last updated on 15/May/19
tan(590°)=tan(360°+180°+50°)=tan(50°)=cot(40°) cos(320°)=cos(360°−40°)=cos(40°) sin(111°)=sin(90°+21°)=cos(21°) cos(159°)=cos(180°−21°)=−cos(21°) cos(279°)=cos(270°+9°)=sin(9°) sin(549°)=sin(360°+180°+9°)=−sin(9°) cot(950°)=cot(1080°−180°+50°)=cot(50°)=tan(40°) sin(400°)=sin(360°+40°)=sin(40°) (((tg^2 (590°))/(cos^2 (320°)))+((sin(111°))/(cos(159°))))(((cos(279°))/(sin(549°)))+((ctg^2 (950°))/(sin^2 (400°)))) =(((cot^2 (40°))/(cos^2 (40°)))+((cos(21°))/(−cos(21°))))(((sin(9°))/(−sin(9°)))+((tan^2 (40°))/(sin^2 (40°)))) =((1/(sin^2 (40°)))−1)((1/(cos^2 (40°)))−1) =(1/(sin^2 (40°)cos^2 (40°)))−(1/(sin^2 (40°)))−(1/(cos^2 (40°)))+1 =(1/(sin^2 (40°)cos^2 (40°)))−((sin^2 40°+cos^2 40°)/(sin^2 (40°)cos^2 (40°)))+1 =(1/(sin^2 (40°)cos^2 (40°)))−(1/(sin^2 (40°)cos^2 (40°)))+1 =1
$${tan}\left(\mathrm{590}°\right)={tan}\left(\mathrm{360}°+\mathrm{180}°+\mathrm{50}°\right)={tan}\left(\mathrm{50}°\right)={cot}\left(\mathrm{40}°\right) \\ $$$${cos}\left(\mathrm{320}°\right)={cos}\left(\mathrm{360}°−\mathrm{40}°\right)={cos}\left(\mathrm{40}°\right) \\ $$$${sin}\left(\mathrm{111}°\right)={sin}\left(\mathrm{90}°+\mathrm{21}°\right)={cos}\left(\mathrm{21}°\right) \\ $$$${cos}\left(\mathrm{159}°\right)={cos}\left(\mathrm{180}°−\mathrm{21}°\right)=−{cos}\left(\mathrm{21}°\right) \\ $$$${cos}\left(\mathrm{279}°\right)={cos}\left(\mathrm{270}°+\mathrm{9}°\right)={sin}\left(\mathrm{9}°\right) \\ $$$${sin}\left(\mathrm{549}°\right)={sin}\left(\mathrm{360}°+\mathrm{180}°+\mathrm{9}°\right)=−{sin}\left(\mathrm{9}°\right) \\ $$$${cot}\left(\mathrm{950}°\right)={cot}\left(\mathrm{1080}°−\mathrm{180}°+\mathrm{50}°\right)={cot}\left(\mathrm{50}°\right)={tan}\left(\mathrm{40}°\right) \\ $$$${sin}\left(\mathrm{400}°\right)={sin}\left(\mathrm{360}°+\mathrm{40}°\right)={sin}\left(\mathrm{40}°\right) \\ $$$$\left(\frac{\boldsymbol{\mathrm{tg}}^{\mathrm{2}} \left(\mathrm{590}°\right)}{\boldsymbol{\mathrm{cos}}^{\mathrm{2}} \left(\mathrm{320}°\right)}+\frac{\boldsymbol{\mathrm{sin}}\left(\mathrm{111}°\right)}{\boldsymbol{\mathrm{cos}}\left(\mathrm{159}°\right)}\right)\left(\frac{\boldsymbol{\mathrm{cos}}\left(\mathrm{279}°\right)}{\boldsymbol{\mathrm{sin}}\left(\mathrm{549}°\right)}+\frac{\boldsymbol{\mathrm{ctg}}^{\mathrm{2}} \left(\mathrm{950}°\right)}{\boldsymbol{\mathrm{sin}}^{\mathrm{2}} \left(\mathrm{400}°\right)}\right) \\ $$$$=\left(\frac{{cot}^{\mathrm{2}} \left(\mathrm{40}°\right)}{{cos}^{\mathrm{2}} \left(\mathrm{40}°\right)}+\frac{{cos}\left(\mathrm{21}°\right)}{−{cos}\left(\mathrm{21}°\right)}\right)\left(\frac{{sin}\left(\mathrm{9}°\right)}{−{sin}\left(\mathrm{9}°\right)}+\frac{{tan}^{\mathrm{2}} \left(\mathrm{40}°\right)}{{sin}^{\mathrm{2}} \left(\mathrm{40}°\right)}\right) \\ $$$$=\left(\frac{\mathrm{1}}{{sin}^{\mathrm{2}} \left(\mathrm{40}°\right)}−\mathrm{1}\right)\left(\frac{\mathrm{1}}{{cos}^{\mathrm{2}} \left(\mathrm{40}°\right)}−\mathrm{1}\right) \\ $$$$=\frac{\mathrm{1}}{{sin}^{\mathrm{2}} \left(\mathrm{40}°\right){cos}^{\mathrm{2}} \left(\mathrm{40}°\right)}−\frac{\mathrm{1}}{{sin}^{\mathrm{2}} \left(\mathrm{40}°\right)}−\frac{\mathrm{1}}{{cos}^{\mathrm{2}} \left(\mathrm{40}°\right)}+\mathrm{1} \\ $$$$=\frac{\mathrm{1}}{{sin}^{\mathrm{2}} \left(\mathrm{40}°\right){cos}^{\mathrm{2}} \left(\mathrm{40}°\right)}−\frac{{sin}^{\mathrm{2}} \mathrm{40}°+{cos}^{\mathrm{2}} \mathrm{40}°}{{sin}^{\mathrm{2}} \left(\mathrm{40}°\right){cos}^{\mathrm{2}} \left(\mathrm{40}°\right)}+\mathrm{1} \\ $$$$=\frac{\mathrm{1}}{{sin}^{\mathrm{2}} \left(\mathrm{40}°\right){cos}^{\mathrm{2}} \left(\mathrm{40}°\right)}−\frac{\mathrm{1}}{{sin}^{\mathrm{2}} \left(\mathrm{40}°\right){cos}^{\mathrm{2}} \left(\mathrm{40}°\right)}+\mathrm{1} \\ $$$$=\mathrm{1} \\ $$$$ \\ $$

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