# (tan alpha+cot alpha)/(sin alpha-cos alpha)=sec^2 alpha+csc^2 alpha

$\frac{\mathrm{tan}\alpha +\mathrm{cot}\alpha }{\mathrm{sin}\alpha -\mathrm{cos}\alpha }={\mathrm{sec}}^{2}\alpha +{\mathrm{csc}}^{2}\alpha$

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Work on the left side. Separate as:

Work on the left side.
Separate as:
$\frac{\mathrm{tan}\alpha +\mathrm{cot}\alpha }{\mathrm{sin}\alpha \cdot \mathrm{cos}\alpha }=\left(\frac{\mathrm{tan}}{\alpha }\mathrm{sin}\alpha \cdot \mathrm{cos}\alpha +\left(\frac{\mathrm{cot}}{\alpha }\mathrm{sin}\alpha \cdot \mathrm{cos}\alpha \right)\right)$
Use the quotient identity for tangent and cotangent:
$\frac{\mathrm{tan}\alpha +\mathrm{cos}\alpha }{\mathrm{sin}\alpha \cdot \mathrm{cos}\alpha }=\left(\left(\frac{\frac{\mathrm{sin}}{\alpha }\mathrm{cos}\alpha }{\mathrm{sin}\alpha \cdot \mathrm{cos}\alpha }\right)+\left(\frac{\frac{\mathrm{cos}}{\alpha }\mathrm{sin}\alpha }{\mathrm{sin}\alpha \cdot \mathrm{cos}\alpha }\right)\right)$
Simplify:
$\left(\frac{\mathrm{tan}\alpha +\mathrm{cot}\alpha }{\mathrm{sin}\alpha }\cdot \mathrm{cos}\alpha \right)=\frac{1}{{\mathrm{cos}}^{2}\alpha }+\frac{1}{{\mathrm{sin}}^{2}\alpha }$
Use the reciprocal identities for cosine and sine:
$\frac{\mathrm{tan}\alpha +\mathrm{cot}\alpha }{\mathrm{sin}\alpha \cdot \mathrm{cos}\alpha }={\mathrm{sec}}^{2}\alpha +{\mathrm{csc}}^{2}\alpha$