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Question Number 27296 by julli deswal last updated on 04/Jan/18

((sin^2 3A)/(sin^2 A)) − ((cos^2 3A)/(cos^2 A)) =

$$\frac{\mathrm{sin}^{\mathrm{2}} \mathrm{3}{A}}{\mathrm{sin}^{\mathrm{2}} {A}}\:−\:\frac{\mathrm{cos}^{\mathrm{2}} \mathrm{3}{A}}{\mathrm{cos}^{\mathrm{2}} {A}}\:=\: \\ $$

Commented by abdo imad last updated on 04/Jan/18

=(((sin(3a)cosa)^2 −(cos(3a)sina)^2 )/((sina cosa)^2 )) =(((sin(3a)cosa −cos(3a)sina)(sin(3a)cosa +cos(3a)sina))/((sina cosa)^2 )) =((sin(2a).sin(4a))/(((1/2)sin(2a))^2 )) =4((sin(4a))/(sin(2a))) =((8sin(2a)cos(2a))/(sin(2a)))= 8cos(2a) .

$$=\frac{\left({sin}\left(\mathrm{3}{a}\right){cosa}\right)^{\mathrm{2}} −\left({cos}\left(\mathrm{3}{a}\right){sina}\right)^{\mathrm{2}} }{\left({sina}\:{cosa}\right)^{\mathrm{2}} } \\ $$$$=\frac{\left({sin}\left(\mathrm{3}{a}\right){cosa}\:−{cos}\left(\mathrm{3}{a}\right){sina}\right)\left({sin}\left(\mathrm{3}{a}\right){cosa}\:+{cos}\left(\mathrm{3}{a}\right){sina}\right)}{\left({sina}\:{cosa}\right)^{\mathrm{2}} } \\ $$$$=\frac{{sin}\left(\mathrm{2}{a}\right).{sin}\left(\mathrm{4}{a}\right)}{\left(\frac{\mathrm{1}}{\mathrm{2}}{sin}\left(\mathrm{2}{a}\right)\right)^{\mathrm{2}} } \\ $$$$=\mathrm{4}\frac{{sin}\left(\mathrm{4}{a}\right)}{{sin}\left(\mathrm{2}{a}\right)}\:=\frac{\mathrm{8}{sin}\left(\mathrm{2}{a}\right){cos}\left(\mathrm{2}{a}\right)}{{sin}\left(\mathrm{2}{a}\right)}=\:\mathrm{8}{cos}\left(\mathrm{2}{a}\right)\:. \\ $$

Answered by Giannibo last updated on 04/Jan/18

((sin^2 (2A+A))/(sin^2 A))−((cos^2 (2A+A))/(cos^2 A))= (((sin 2A∙cos A+cos 2A∙sin A)^2 )/(sin^2 A))−(((cos 2A∙cos A−sin 2A∙sin A)^2 )/(cos^2 A)) (((2sin A∙cos^2 A+(cos^2 A−sin^2 A)sin A)^2 )/(sin^2 A))−((((cos^2 A−sin^2 A)cos A−2sin A∙cos A∙sin A)^2 )/(cos^2 A))= ((sin^2 A(2cos^2 A+cos^2 A−sin^2 A)^2 )/(sin^2 A))−((cos^2 A∙(cos^2 A−sin^2 A−2sin^2 A)^2 )/(cos^2 A))= (3cos^2 A−sin^2 A)^2 −(cos^2 A−3sin^2 A)^2 = 9cos^4 A−6cos^2 Asin^2 A+sin^4 A−cos^4 A+6cos^2 Asin^2 A−9sin^4 A= 8cos^4 A−8sin^4 A= 8(cos^2 A−sin^2 A)(cos^2 A+sin^2 A)= 8cos 2A

$$ \\ $$$$ \\ $$$$\frac{\mathrm{sin}^{\mathrm{2}} \left(\mathrm{2A}+\mathrm{A}\right)}{\mathrm{sin}^{\mathrm{2}} \:\mathrm{A}}−\frac{\mathrm{cos}\:^{\mathrm{2}} \left(\mathrm{2A}+\mathrm{A}\right)}{\mathrm{cos}\:^{\mathrm{2}} \mathrm{A}}= \\ $$$$\frac{\left(\mathrm{sin}\:\mathrm{2A}\centerdot\mathrm{cos}\:\mathrm{A}+\mathrm{cos}\:\mathrm{2A}\centerdot\mathrm{sin}\:\mathrm{A}\right)^{\mathrm{2}} }{\mathrm{sin}\:^{\mathrm{2}} \mathrm{A}}−\frac{\left(\mathrm{cos}\:\mathrm{2A}\centerdot\mathrm{cos}\:\mathrm{A}−\mathrm{sin}\:\mathrm{2A}\centerdot\mathrm{sin}\:\mathrm{A}\right)^{\mathrm{2}} \:}{\mathrm{cos}\:^{\mathrm{2}} \mathrm{A}} \\ $$$$\frac{\left(\mathrm{2sin}\:\mathrm{A}\centerdot\mathrm{cos}^{\mathrm{2}} \:\mathrm{A}+\left(\mathrm{cos}\:^{\mathrm{2}} \mathrm{A}−\mathrm{sin}\:^{\mathrm{2}} \mathrm{A}\right)\mathrm{sin}\:\mathrm{A}\right)^{\mathrm{2}} }{\mathrm{sin}\:^{\mathrm{2}} \mathrm{A}}−\frac{\left(\left(\mathrm{cos}\:^{\mathrm{2}} \mathrm{A}−\mathrm{sin}\:^{\mathrm{2}} \mathrm{A}\right)\mathrm{cos}\:\mathrm{A}−\mathrm{2sin}\:\mathrm{A}\centerdot\mathrm{cos}\:\mathrm{A}\centerdot\mathrm{sin}\:\mathrm{A}\right)^{\mathrm{2}} }{\mathrm{cos}\:^{\mathrm{2}} \mathrm{A}}= \\ $$$$\frac{\mathrm{sin}\:^{\mathrm{2}} \mathrm{A}\left(\mathrm{2cos}^{\mathrm{2}} \:\mathrm{A}+\mathrm{cos}\:^{\mathrm{2}} \mathrm{A}−\mathrm{sin}\:^{\mathrm{2}} \mathrm{A}\right)^{\mathrm{2}} }{\mathrm{sin}\:^{\mathrm{2}} \mathrm{A}}−\frac{\mathrm{cos}\:^{\mathrm{2}} \mathrm{A}\centerdot\left(\mathrm{cos}\:^{\mathrm{2}} \mathrm{A}−\mathrm{sin}\:^{\mathrm{2}} \mathrm{A}−\mathrm{2sin}\:^{\mathrm{2}} \mathrm{A}\right)^{\mathrm{2}} }{\mathrm{cos}\:^{\mathrm{2}} \mathrm{A}}= \\ $$$$\left(\mathrm{3cos}\:^{\mathrm{2}} \mathrm{A}−\mathrm{sin}\:^{\mathrm{2}} \mathrm{A}\right)^{\mathrm{2}} −\left(\mathrm{cos}\:^{\mathrm{2}} \mathrm{A}−\mathrm{3sin}\:^{\mathrm{2}} \mathrm{A}\right)^{\mathrm{2}} = \\ $$$$\mathrm{9cos}\:^{\mathrm{4}} \mathrm{A}−\mathrm{6cos}\:^{\mathrm{2}} \mathrm{Asin}\:^{\mathrm{2}} \mathrm{A}+\mathrm{sin}\:^{\mathrm{4}} \mathrm{A}−\mathrm{cos}\:^{\mathrm{4}} \mathrm{A}+\mathrm{6cos}\:^{\mathrm{2}} \mathrm{Asin}\:^{\mathrm{2}} \mathrm{A}−\mathrm{9sin}\:^{\mathrm{4}} \mathrm{A}= \\ $$$$\mathrm{8cos}\:^{\mathrm{4}} \mathrm{A}−\mathrm{8sin}\:^{\mathrm{4}} \mathrm{A}= \\ $$$$\mathrm{8}\left(\mathrm{cos}\:^{\mathrm{2}} \mathrm{A}−\mathrm{sin}\:^{\mathrm{2}} \mathrm{A}\right)\left(\mathrm{cos}\:^{\mathrm{2}} \mathrm{A}+\mathrm{sin}\:^{\mathrm{2}} \mathrm{A}\right)= \\ $$$$\mathrm{8cos}\:\mathrm{2A} \\ $$

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