# Solve the equation frac{sin^{2}theta/}{cos theta}= sec theta-cos theta

Solve the equation $\frac{{\mathrm{sin}}^{2}\theta /}{cos\theta }=\mathrm{sec}\theta -\mathrm{cos}\theta$
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Roosevelt Houghton
Work with the left side. Use the Pythagorean identity:
${\mathrm{sin}}^{2}\theta +{\mathrm{cos}}^{2}\theta =1$
$\frac{{\mathrm{sin}}^{2}\theta }{cos\theta }=\frac{1-{\mathrm{cos}}^{2}\theta }{cos\theta }$
Separate as:
$\frac{{\mathrm{sin}}^{2}\theta }{cos\theta }=1/\mathrm{cos}\theta -\frac{{\mathrm{cos}}^{2}\theta }{cos\theta }$
$\frac{{\mathrm{sin}}^{2}\theta }{\mathrm{cos}\theta }=\frac{1}{\mathrm{cos}\theta }-\mathrm{cos}\theta$
Use the reciptiocal identity: $\mathrm{sec}\theta =\frac{1}{\mathrm{cos}\theta }$
$\frac{{\mathrm{sin}}^{2}\theta }{\mathrm{cos}\theta }=\mathrm{sec}\theta -\mathrm{cos}\theta$