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

Step-by-step Solution

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Final answer to the problem

true

Step-by-step Solution

How should I solve this problem?

  • Dimostrare dal LHS (lato sinistro)
  • Dimostrare da RHS (lato destro)
  • Esprimere tutto in seno e coseno
  • Equazione differenziale esatta
  • Equazione differenziale lineare
  • Equazione differenziale separabile
  • Equazione differenziale omogenea
  • Prodotto di binomi con termine comune
  • Metodo FOIL
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Can't find a method? Tell us so we can add it.
1

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

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

Apply the trigonometric identity: $\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

Applying the trigonometric identity: $\cot\left(\theta \right) = \frac{\cos\left(\theta \right)}{\sin\left(\theta \right)}$

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

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

Apply the trigonometric identity: $\frac{n}{\cos\left(\theta \right)}$$=n\sec\left(\theta \right)$, where $n=1$

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

Apply the trigonometric identity: $\frac{n}{\sin\left(\theta \right)}$$=n\csc\left(\theta \right)$, where $n=\sec\left(x\right)$

$\sec\left(x\right)\csc\left(x\right)$
9

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

true

Final answer to the problem

true

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