Prove that sec(theta)+csc(theta)=(sin(theta)+cos(theta))(tan(theta)+cot(theta))

mattgondek4 2021-01-07 Answered
Prove that $\mathrm{sec}\left(\theta \right)+\mathrm{csc}\left(\theta \right)=\left(\mathrm{sin}\left(\theta \right)+\mathrm{cos}\left(\theta \right)\right)\left(\mathrm{tan}\left(\theta \right)+\mathrm{cot}\left(\theta \right)\right)$
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Expert Answer

SchepperJ
Answered 2021-01-08 Author has 96 answers
RHS $=\left(\mathrm{sin}\left(t\right)+\mathrm{cos}\left(t\right)\right)\cdot \left(\mathrm{tan}\left(t\right)+\mathrm{cot}\left(t\right)\right)=\left(\mathrm{sin}\left(t\right)+\mathrm{cos}\left(t\right)\right)\cdot \left(\frac{\mathrm{sin}\left(t\right)}{\mathrm{cos}\left(t\right)}+\frac{\mathrm{cos}\left(t\right)}{\mathrm{sin}\left(t\right)}\right)=\left(\mathrm{sin}\left(t\right)+\mathrm{cos}\left(t\right)\right)\cdot \frac{{\mathrm{sin}}^{2}\left(t\right)+{\mathrm{cos}}^{2}\left(t\right)}{\left(\mathrm{sin}\left(t\right).\mathrm{cos}\left(t\right)\right)}$
RHS $=\left(\mathrm{sin}\left(t\right)+\mathrm{cos}\left(t\right)\right)\cdot \frac{1}{\left(\mathrm{sin}\left(t\right).\mathrm{cos}\left(t\right)\right)=\left(\frac{1}{\mathrm{cos}\left(t\right)}+\frac{1}{\mathrm{sin}\left(t\right)}\right)=\mathrm{sec}\left(t\right)+\mathrm{csc}\left(t\right)=}$ LHS
RHS = LHS
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