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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?

• Prove from RHS (right-hand side)
• Prove from LHS (left-hand side)
• Express everything into Sine and Cosine
• 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.
1

Starting from the right-hand side (RHS) of the identity

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

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

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

Applying the cosecant identity: $\displaystyle\csc\left(\theta\right)=\frac{1}{\sin\left(\theta\right)}$

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

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

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

Multiply $\frac{1}{\cos\left(x\right)\sin\left(x\right)}$ by $\frac{sin(x)^2+cos(x)^2}{sin(x)^2+cos(x)^2}$

$\frac{1}{\cos\left(x\right)\sin\left(x\right)}\frac{\sin\left(x\right)^2+\cos\left(x\right)^2}{\sin\left(x\right)^2+\cos\left(x\right)^2}$
6

Multiplying fractions $\frac{1}{\cos\left(x\right)\sin\left(x\right)} \times \frac{\sin\left(x\right)^2+\cos\left(x\right)^2}{\sin\left(x\right)^2+\cos\left(x\right)^2}$

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

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

$\frac{\sin\left(x\right)^2+\cos\left(x\right)^2}{\cos\left(x\right)\sin\left(x\right)}$
Why is sin(x)^2 + cos(x)^2 = 1 ?
8

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 ?
9

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

true

true

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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.