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

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true

##  Step-by-step Solution 

How should I solve this problem?

• Express everything into Sine and Cosine
• Prove from LHS (left-hand side)
• Prove from RHS (right-hand side)
• Exact Differential Equation
• Linear Differential Equation
• Separable Differential Equation
• Homogeneous Differential Equation
• Integrate by partial fractions
• Product of Binomials with Common Term
• FOIL Method
Can't find a method? Tell us so we can add it.

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

1

Start from the LHS (left-hand side)

$\tan\left(x\right)+\cot\left(x\right)$
2

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

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

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

$\frac{\sin\left(x\right)}{\cos\left(x\right)}+\frac{\cos\left(x\right)}{\sin\left(x\right)}$
Why is cot(x) = cos(x)/sin(x) ?
4

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

$\frac{\sin\left(x\right)^2+\cos\left(x\right)\cos\left(x\right)}{\cos\left(x\right)\sin\left(x\right)}$
6

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

$\frac{\sin\left(x\right)^2+\cos\left(x\right)^2}{\cos\left(x\right)\sin\left(x\right)}$
7

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

$\frac{1}{\cos\left(x\right)\sin\left(x\right)}$
Why is sin(x)^2 + cos(x)^2 = 1 ?

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

$\frac{1}{\cos\left(x\right)}\csc\left(x\right)$
10

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

$\frac{1}{\cos\left(x\right)}\frac{1}{\sin\left(x\right)}$
11

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 right side $\frac{1}{\cos\left(x\right)\sin\left(x\right)}$ to reach the left 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 we have reached the expression of our goal, we have proven the identity

true

true

##  Explore different ways to solve this problem

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###  Main Topic: Trigonometric Identities

In mathematics, trigonometric identities are equalities that involve trigonometric functions and are true for every single value of the occurring variables where both sides of the equality are defined.