# (cos​233-cos257)/(sin 21-cos 21)=?

Dear Student,
$\frac{{\mathrm{cos}}^{2}33-{\mathrm{cos}}^{2}57}{\mathrm{sin}21-\mathrm{cos}21}\phantom{\rule{0ex}{0ex}}=\frac{{\mathrm{sin}}^{2}\left(90-33\right)-{\mathrm{sin}}^{2}\left(90-57\right)}{\mathrm{sin}21-\mathrm{sin}\left(90-21\right)}\phantom{\rule{0ex}{0ex}}=\frac{{\mathrm{sin}}^{2}\left(57\right)-{\mathrm{sin}}^{2}\left(33\right)}{\mathrm{sin}21-\mathrm{sin}69}\phantom{\rule{0ex}{0ex}}=\frac{\mathrm{sin}\left(57+33\right).\mathrm{sin}\left(57-33\right)}{2\mathrm{cos}\left(\frac{21+69}{2}\right)\mathrm{sin}\left(\frac{21-69}{2}\right)}\phantom{\rule{0ex}{0ex}}=\frac{\mathrm{sin}\left(90\right).\mathrm{sin}\left(24\right)}{2\mathrm{cos}\left(45\right)\mathrm{sin}\left(-24\right)}\phantom{\rule{0ex}{0ex}}=-\frac{\mathrm{sin}\left(90\right).\mathrm{sin}\left(24\right)}{2\mathrm{cos}\left(45\right)\mathrm{sin}\left(24\right)}\phantom{\rule{0ex}{0ex}}=-\frac{\mathrm{sin}\left(90\right)}{2\mathrm{cos}\left(45\right)}\phantom{\rule{0ex}{0ex}}=-\frac{1}{2×\frac{1}{\sqrt{2}}}\phantom{\rule{0ex}{0ex}}=-\frac{1}{\sqrt{2}}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}$