Simplify the trigonometric expression (-sec(x))/tan(x)

 Exercise

$\frac{-secx}{tanx}$

 

Applying the secant identity: $\displaystyle\sec\left(\theta\right)=\frac{1}{\cos\left(\theta\right)}$

$\frac{\frac{-1}{\cos\left(x\right)}}{\tan\left(x\right)}$

 

Applying the tangent identity: $\displaystyle\tan\left(\theta\right)=\frac{\sin\left(\theta\right)}{\cos\left(\theta\right)}$

$\frac{\frac{-1}{\cos\left(x\right)}}{\frac{\sin\left(x\right)}{\cos\left(x\right)}}$

 Why is tan(x) = sin(x)/cos(x) ?

 

Simplify the fraction $\frac{\frac{-1}{\cos\left(x\right)}}{\frac{\sin\left(x\right)}{\cos\left(x\right)}}$

$\frac{-1}{\sin\left(x\right)}$

 

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

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

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

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