babeeb0oL

2021-02-09

To find the equation $\left(1-\mathrm{sin}\beta \right)\left(1+\mathrm{csc}\beta \right)=\mathrm{csc}\beta -\mathrm{sin}\beta$

Work on the left side. Apply distributive property:

$\left(1-\mathrm{sin}\beta \right)\left(1+\mathrm{csc}\beta \right)=1\left(1-\mathrm{sin}\beta \right)+\mathrm{csc}\beta \left(1-\mathrm{sin}\beta \right)$

$\left(1-\mathrm{sin}\beta \right)\left(1+\mathrm{csc}\beta \right)=1-\mathrm{sin}\beta +\mathrm{csc}\beta -\mathrm{sin}\beta \mathrm{csc}\beta$

Use the reciprocal identity for cosecant:

$\left(1-\mathrm{sin}\beta \right)\left(1+\mathrm{csc}\beta \right)=1-\mathrm{sin}\beta +\mathrm{csc}\beta -\mathrm{sin}\beta ×1\mathrm{sin}\beta$

Simplify:

$\left(1-\mathrm{sin}\beta \right)\left(1+\mathrm{csc}\beta \right)=1-\mathrm{sin}\beta +\mathrm{csc}\beta -1$

$\left(1-\mathrm{sin}\beta \right)\left(1+\mathrm{csc}\beta \right)=-\mathrm{sin}\beta +\mathrm{csc}\beta$

or

$\left(1-\mathrm{sin}\beta \right)\left(1+\mathrm{csc}\beta \right)=\mathrm{csc}\beta -\mathrm{sin}\beta$

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