# Solve. tan 1/2x=csc x-cot x

Solve.
$\mathrm{tan}1/2x=\mathrm{csc}x-\mathrm{cot}x$
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Alden Holder
$\left({\mathrm{cos}}^{2}x{\right)}^{2}+1-\left({\mathrm{sin}}^{2}x{\right)}^{2}=2{\mathrm{cos}}^{2}x$
identity: ${\mathrm{cos}}^{2}x+{\mathrm{sin}}^{2}x=1so{\mathrm{sin}}^{2}x=1-{\mathrm{cos}}^{2}x$
$\left({\mathrm{cos}}^{2}x{\right)}^{2}+1-\left(1-{\mathrm{cos}}^{2}x{\right)}^{2}=2{\mathrm{cos}}^{2}x$
$\left({\mathrm{cos}}^{2}x{\right)}^{2}+1-\left[1-2{\mathrm{cos}}^{2}x+{\mathrm{cos}}^{4}x\right]=2{\mathrm{cos}}^{2}x$
${\mathrm{cos}}^{4}x+1-1+2{\mathrm{cos}}^{2}x-{\mathrm{cos}}^{4}x=2{\mathrm{cos}}^{2}x$
Cancel: $2{\mathrm{cos}}^{2}x=2{\mathrm{cos}}^{2}x$
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Ishaan Booker
${\mathrm{cos}}^{4}x+1-\left(1-{\mathrm{cos}}^{2}x\right)2=2{\mathrm{cos}}^{2}x,{\mathrm{cos}}^{4}x+1-\left(1+{\mathrm{cos}}^{4}x-2{\mathrm{cos}}^{2}x\right)=2{\mathrm{cos}}^{2}x$