 # Solve the equation csc x - sin x = cot x cos x Marvin Mccormick 2021-01-17 Answered
Solve the equation $\mathrm{csc}x-\mathrm{sin}x=\mathrm{cot}x\mathrm{cos}x$
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Work u\sing the left side
Use the reciptional identity: $\mathrm{csc}x=\frac{1}{\mathrm{sin}x}$
$\mathrm{csc}x-\mathrm{sin}x=\frac{1}{\mathrm{sin}x-\mathrm{sin}x}$
Simplify the right side into a \single expression:
$\mathrm{csc}x-\mathrm{sin}x=\left(1-{\mathrm{sin}}^{2}x\right)-\mathrm{sin}x$
Use the Pythagorean identity: ${\mathrm{sin}}^{2}x+{\mathrm{cos}}^{2}x=1$
$\mathrm{csc}x-\mathrm{sin}x={\mathrm{cos}}^{2}x\mathrm{sin}x$
Separate as:
$\mathrm{csc}x-\mathrm{sin}x=\frac{\mathrm{cos}x}{\mathrm{sin}x}×\mathrm{cos}x$
Use the quotient identity: $\mathrm{cot}x=\frac{\mathrm{cos}x}{\mathrm{sin}x}$
$\mathrm{csc}-\mathrm{sin}x=\mathrm{cot}x\mathrm{cos}x$