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Question Number 113488 by weltr last updated on 13/Sep/20

prove that  ((sin(2a)cos(2a))/(cos(4a))) = (1/2) tan(4a)

$${prove}\:{that} \\ $$$$\frac{{sin}\left(\mathrm{2}{a}\right){cos}\left(\mathrm{2}{a}\right)}{{cos}\left(\mathrm{4}{a}\right)}\:=\:\frac{\mathrm{1}}{\mathrm{2}}\:{tan}\left(\mathrm{4}{a}\right) \\ $$

Answered by som(math1967) last updated on 13/Sep/20

L.H.S=((2sin2αcos2α)/(2cos4α))  =(1/2)×((sin4α)/(cos4α))=(1/2)tan4α=R.H.S

$$\mathrm{L}.\mathrm{H}.\mathrm{S}=\frac{\mathrm{2sin2}\alpha\mathrm{cos2}\alpha}{\mathrm{2cos4}\alpha} \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}×\frac{\mathrm{sin4}\alpha}{\mathrm{cos4}\alpha}=\frac{\mathrm{1}}{\mathrm{2}}\mathrm{tan4}\alpha=\mathrm{R}.\mathrm{H}.\mathrm{S} \\ $$

Commented by weltr last updated on 13/Sep/20

thanks

$${thanks} \\ $$

Answered by floor(10²Eta[1]) last updated on 13/Sep/20

((sin(2a)cos(2a))/(cos(4a)))=((2sin(2a)cos(2a))/(2cos(4a)))  =((sin(4a))/(2cos(4a)))=(1/2)tan(4a)

$$\frac{\mathrm{sin}\left(\mathrm{2a}\right)\mathrm{cos}\left(\mathrm{2a}\right)}{\mathrm{cos}\left(\mathrm{4a}\right)}=\frac{\mathrm{2sin}\left(\mathrm{2a}\right)\mathrm{cos}\left(\mathrm{2a}\right)}{\mathrm{2cos}\left(\mathrm{4a}\right)} \\ $$$$=\frac{\mathrm{sin}\left(\mathrm{4a}\right)}{\mathrm{2cos}\left(\mathrm{4a}\right)}=\frac{\mathrm{1}}{\mathrm{2}}\mathrm{tan}\left(\mathrm{4a}\right) \\ $$$$ \\ $$

Commented by weltr last updated on 13/Sep/20

thanks

$${thanks} \\ $$

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