# sin-x-cos-x-a-pi-4-x-pi-2-Which-statement-is-correct-1-sin-2-x-cos-2-x-1-2-3-2a-2-a-4-2-sin-4-x-cos-4-x-1-8-3a-4-6a-2-5-3-sin-2x-1-a-2-2-

Question Number 11456 by Joel576 last updated on 26/Mar/17
$$\mathrm{sin}\:{x}\:−\:\mathrm{cos}\:{x}\:=\:{a},\:\:\frac{\pi}{\mathrm{4}}\:\leqslant\:{x}\:\leqslant\:\frac{\pi}{\mathrm{2}} \\$$$$\mathrm{Which}\:\mathrm{statement}\:\mathrm{is}\:\mathrm{correct}? \\$$$$\left(\mathrm{1}\right)\:\mathrm{sin}^{\mathrm{2}} \:{x}\:−\:\mathrm{cos}^{\mathrm{2}} \:{x}\:=\:−\frac{\mathrm{1}}{\mathrm{2}}\sqrt{\mathrm{3}\:+\:\mathrm{2}{a}^{\mathrm{2}} \:−\:{a}^{\mathrm{4}} } \\$$$$\left(\mathrm{2}\right)\:\mathrm{sin}^{\mathrm{4}} \:{x}\:+\:\mathrm{cos}^{\mathrm{4}} \:{x}\:=\:\frac{\mathrm{1}}{\mathrm{8}}\left(−\mathrm{3}{a}^{\mathrm{4}} \:+\:\mathrm{6}{a}^{\mathrm{2}} \:+\:\mathrm{5}\right) \\$$$$\left(\mathrm{3}\right)\:\mathrm{sin}\:\mathrm{2}{x}\:=\:\frac{\mathrm{1}\:−\:{a}^{\mathrm{2}} }{\mathrm{2}} \\$$$$\left(\mathrm{4}\right)\:\mathrm{tan}^{\mathrm{2}} \:{x}\:+\:\mathrm{cot}^{\mathrm{2}} \:{x}\:=\:\frac{−\mathrm{3}{a}^{\mathrm{4}} \:−\:\mathrm{2}{a}^{\mathrm{2}} \:+\:\mathrm{8}{a}\:+\:\mathrm{13}\:}{\mathrm{2}\left({a}^{\mathrm{4}} \:+\:\mathrm{2}{a}^{\mathrm{2}} \:+\:\mathrm{1}\right)} \\$$
Commented by ajfour last updated on 26/Mar/17
$$\mathrm{none}\:\mathrm{seem}\:\mathrm{to}\:\mathrm{be}\:\mathrm{correct}.. \\$$
Commented by Joel576 last updated on 26/Mar/17
$${why}? \\$$
Commented by Joel576 last updated on 26/Mar/17
$$\mathrm{sorry},\:\mathrm{in}\:\mathrm{statement}\:\left(\mathrm{4}\right)\:\mathrm{it}\:\mathrm{must}\:\mathrm{be}\:\mathrm{tan}^{\mathrm{2}} \:{x}\:+\:\mathrm{cot}^{\mathrm{2}} \:{x} \\$$
Answered by ridwan balatif last updated on 26/Mar/17
Answered by ridwan balatif last updated on 26/Mar/17
Commented by Joel576 last updated on 27/Mar/17
$${thank}\:{you}\:{very}\:{much} \\$$