# Step-by-step Solution

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## Step-by-step explanation

Problem to solve:

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

Choose 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)

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

$\frac{\sin\left(x\right)}{\cos\left(x\right)}+\cot\left(x\right)$
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Rewrite $\cot\left(x\right)$ in terms of sine and cosine

$\frac{\sin\left(x\right)}{\cos\left(x\right)}+\frac{\cos\left(x\right)}{\sin\left(x\right)}$
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Combine fractions with different denominator using the formula: $\displaystyle\frac{a}{b}+\frac{c}{d}=\frac{a\cdot d + b\cdot c}{b\cdot d}$

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

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

$\frac{\sin\left(x\right)^2+\cos\left(x\right)\cos\left(x\right)}{\cos\left(x\right)\sin\left(x\right)}$
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When multiplying two powers that have the same base ($\cos\left(x\right)$), you can add the exponents

$\frac{\sin\left(x\right)^2+\cos\left(x\right)^2}{\cos\left(x\right)\sin\left(x\right)}$
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Applying the pythagorean identity: $\sin^2\left(\theta\right)+\cos^2\left(\theta\right)=1$

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

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

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Start from the RHS (right-hand side)

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

$\frac{1}{\cos\left(x\right)}\csc\left(x\right)$
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Rewrite $\csc\left(x\right)$ in terms of sine and cosine

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

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

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 $\frac{1}{\cos\left(x\right)\sin\left(x\right)}$ to reach the right side $\frac{1}{\cos\left(x\right)\sin\left(x\right)}$

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

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

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

true

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