Question Number 23304 by Tinkutara last updated on 28/Oct/17
$$\mathrm{If}\:\mathrm{sin}^{\mathrm{4}} \mathrm{x}\:+\:\mathrm{cos}^{\mathrm{4}} \mathrm{y}\:+\:\mathrm{2}\:=\:\mathrm{4}\:\mathrm{sinx}.\mathrm{cosy}\:\& \\ $$$$\mathrm{0}\:\leqslant\:{x},{y}\:\leqslant\:\frac{\pi}{\mathrm{2}}\:,\:\mathrm{then}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\mathrm{sin}{x}\:+ \\ $$$$\mathrm{cos}{y}\:\mathrm{is}\:\mathrm{equal}\:\mathrm{to} \\ $$
Commented by Joel577 last updated on 28/Oct/17
$${What}'{s}\:{the}\:{name}\:{of}\:{ur}\:{book}? \\ $$
Commented by Joel577 last updated on 28/Oct/17
$${Owh},\:{I}\:{think}\:{it}\:{was}\:{from}\:{ur}\:{textbook}. \\ $$$${Because}\:{u}\:{have}\:{asked}\:{so}\:{many}\:{questions} \\ $$$${I}\:{wonder}\:{how}\:{many}\:{assignments}\:{did}\:{u}\:{get}\:{from}\:{school} \\ $$$${everyday} \\ $$
Answered by ajfour last updated on 28/Oct/17
$${let}\:\mathrm{sin}\:{x}+\mathrm{cos}\:{y}={S} \\ $$$$\:\:\:\:\:\:\:\mathrm{sin}\:{x}−\mathrm{cos}\:{y}={D} \\ $$$$\:\:\:\:\:\:\:\mathrm{sin}\:{x}\mathrm{cos}\:{y}\:\:=\:{P} \\ $$$${Then}\:{what}\:{is}\:{given}\:{converts}\:{to} \\ $$$$\left(\mathrm{sin}\:^{\mathrm{2}} {x}−\mathrm{cos}\:^{\mathrm{2}} {y}\right)^{\mathrm{2}} +\mathrm{2sin}\:^{\mathrm{2}} {x}\mathrm{cos}\:^{\mathrm{2}} {y}+\mathrm{2} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\:\mathrm{4sin}\:{x}\mathrm{cos}\:{y} \\ $$$$\Rightarrow\:\:{S}^{\:\:\mathrm{2}} {D}^{\mathrm{2}} +\mathrm{2}{P}^{\:\mathrm{2}} +\mathrm{2}−\mathrm{4}{P}=\mathrm{0} \\ $$$$\Rightarrow\:\:{S}^{\:\:\mathrm{2}} {D}^{\mathrm{2}} +\mathrm{2}\left({P}−\mathrm{1}\right)^{\mathrm{2}} =\mathrm{0} \\ $$$$\:{S}\:{cannot}\:{be}\:{zero}\:{for}\:\:\mathrm{0}\leqslant{x},\:{y}\:\leqslant\frac{\pi}{\mathrm{2}} \\ $$$${So}\:\:{D}=\mathrm{0}\:\:{and}\:{P}\:=\mathrm{1} \\ $$$$\Rightarrow\mathrm{sin}\:{x}=\mathrm{cos}\:{y}\:\:{and}\:\mathrm{sin}\:{x}\mathrm{cos}\:{y}=\mathrm{1} \\ $$$${this}\:{can}\:{be}\:{true}\:{only}\:{if} \\ $$$$\:\:\:{x}=\frac{\pi}{\mathrm{2}}\:\:{and}\:{y}=\mathrm{0} \\ $$$${So}\:\:\mathrm{sin}\:{x}+\mathrm{cos}\:{y}\:=\mathrm{2}\:. \\ $$
Commented by Tinkutara last updated on 29/Oct/17
$$\mathrm{Thank}\:\mathrm{you}\:\mathrm{very}\:\mathrm{much}\:\mathrm{Sir}! \\ $$