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# Prove the trigonometric identity $\cot\left(x\right)\sec\left(x\right)=\csc\left(x\right)$

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## Step-by-step Solution

Problem to solve:

$\cot\left(x\right)\cdot\sec\left(x\right)=\csc\left(x\right)$

Choose the solving method

1

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

$\cot\left(x\right)\frac{1}{\cos\left(x\right)}=\csc\left(x\right)$
2

Apply the trigonometric identity: $\displaystyle\cot(x)=\frac{\cos(x)}{\sin(x)}$

$\frac{\cos\left(x\right)}{\sin\left(x\right)}\frac{1}{\cos\left(x\right)}=\csc\left(x\right)$
3

Multiplying fractions $\frac{\cos\left(x\right)}{\sin\left(x\right)} \times \frac{1}{\cos\left(x\right)}$

$\frac{\cos\left(x\right)}{\sin\left(x\right)\cos\left(x\right)}=\csc\left(x\right)$
4

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

$\frac{1}{\sin\left(x\right)}=\csc\left(x\right)$
5

The reciprocal sine function is cosecant: $\frac{1}{\sin(x)}=\csc(x)$

$\csc\left(x\right)=\csc\left(x\right)$
6

Since both sides of the equality are equal, we have proven the identity

true

true
$\cot\left(x\right)\cdot\sec\left(x\right)=\csc\left(x\right)$

### Main topic:

Trigonometric Identities

~ 0.04 s