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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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##  Step-by-step Solution 

Problem to solve:

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

Specify the solving method

1

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

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

Applying the tangent identity: $\displaystyle\tan\left(\theta\right)=\frac{\sin\left(\theta\right)}{\cos\left(\theta\right)}$

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

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

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

The least common multiple (LCM) of a sum of algebraic fractions consists of the product of the common factors with the greatest exponent, and the uncommon factors

$L.C.M.=\cos\left(x\right)\sin\left(x\right)$
5

Obtained the least common multiple (LCM), we place it as the denominator of each fraction, and in the numerator of each fraction we add the factors that we need to complete

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

Rewrite the sum of fractions as a single fraction with the same denominator

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

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

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

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

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

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 ?
6

Combine and simplify all terms in the same fraction with common denominator $\cos\left(x\right)\sin\left(x\right)$

$\frac{1}{\cos\left(x\right)\sin\left(x\right)}$
7

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

$\frac{1\sec\left(x\right)}{\sin\left(x\right)}$
8

Any expression multiplied by $1$ is equal to itself

$\frac{\sec\left(x\right)}{\sin\left(x\right)}$
9

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

$\sec\left(x\right)\csc\left(x\right)$
10

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

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