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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:

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

Specify the solving method

1

Starting from the left-hand side (LHS) of the identity

$\cot\left(x\right)\sec\left(x\right)$
2

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

$\frac{\cos\left(x\right)}{\sin\left(x\right)}\sec\left(x\right)$
Why does cot(x) = (cos(x)/(sin(x) ?
3

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

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

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

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

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

$\frac{1}{\sin\left(x\right)}$
4

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

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

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

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

Since we have reached the expression of our goal, we have proven the identity

true

true

##  Explore different ways to solve this problem

Solving a math problem using different methods is important because it enhances understanding, encourages critical thinking, allows for multiple solutions, and develops problem-solving strategies. Read more

Prove from RHS (right-hand side)Express everything into Sine and Cosine

### Main topic:

Trigonometric Identities

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