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

Step-by-step Solution

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Final Answer

true

Step-by-step Solution

Problem to solve:

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

Specify the solving method

I. Express the LHS in terms of sine and cosine and simplify

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Start from the LHS (left-hand side)

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

Rewrite $\cot\left(x\right)$ in terms of sine and cosine

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

Rewrite $\sec\left(x\right)$ in terms of sine and cosine

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

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

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

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

II. Express the RHS in terms of sine and cosine and simplify

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Start from the RHS (right-hand side)

$\csc\left(x\right)$
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Rewrite $\csc\left(x\right)$ in terms of sine and cosine

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

III. Choose what side of the identity are we going to work on

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To prove an identity, we usually begin to work on the side of the equality that seems to be more complicated, or the side that is not expressed in terms of sine and cosine. In this problem, we will choose to work on the left side $\frac{1}{\sin\left(x\right)}$ to reach the right side $\frac{1}{\sin\left(x\right)}$

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

IV. Check if we arrived at the expression we wanted to prove

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

true

Final Answer

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 LHS (left-hand side)Prove from RHS (right-hand side)

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