Simplify the trigonometric expression ((csc(x)-sin(x))csc(x))/cot(x)

 Exercise

$\frac{\left(\csc x-\sin x\right)\csc x}{\cot x}$

 

Multiply the single term $\csc\left(x\right)$ by each term of the polynomial $\left(\csc\left(x\right)-\sin\left(x\right)\right)$

$\csc\left(x\right)\csc\left(x\right)-\sin\left(x\right)\csc\left(x\right)$

 

When multiplying two powers that have the same base ($\csc\left(x\right)$), you can add the exponents

$\frac{\csc\left(x\right)^2-\sin\left(x\right)\csc\left(x\right)}{\cot\left(x\right)}$

 

Applying the cosecant identity: $\displaystyle\csc\left(\theta\right)=\frac{1}{\sin\left(\theta\right)}$

$\frac{\csc\left(x\right)^2+\sin\left(x\right)\frac{-1}{\sin\left(x\right)}}{\cot\left(x\right)}$

 

Multiplying the fraction by $\sin\left(x\right)$

$\frac{\csc\left(x\right)^2+\frac{-\sin\left(x\right)}{\sin\left(x\right)}}{\cot\left(x\right)}$

 

Simplify the fraction $\frac{-\sin\left(x\right)}{\sin\left(x\right)}$ by $\sin\left(x\right)$

$\frac{\csc\left(x\right)^2-1}{\cot\left(x\right)}$

$\frac{\csc\left(x\right)^2-1}{\cot\left(x\right)}$

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