Final answer to the problem
Step-by-step Solution
How should I solve this problem?
- Exprimez tout en sinus et en cosinus
- Prouver à partir du LHS (côté gauche)
- Prouver à partir du RHS (côté droit)
- Equation différentielle exacte
- Équation différentielle linéaire
- Équation différentielle séparable
- Equation différentielle homogène
- Produit de binômes avec terme commun
- Méthode FOIL
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I. Express the LHS in terms of sine and cosine and simplify
Start from the LHS (left-hand side)
Apply the trigonometric identity: $\tan\left(\theta \right)$$=\frac{\sin\left(\theta \right)}{\cos\left(\theta \right)}$
Apply the trigonometric identity: $\cot\left(\theta \right)$$=\frac{\cos\left(\theta \right)}{\sin\left(\theta \right)}$
Apply the formula: $\frac{a}{b}+\frac{c}{f}$$=\frac{af+cb}{bf}$, where $a=\sin\left(x\right)$, $b=\cos\left(x\right)$, $c=\cos\left(x\right)$ and $f=\sin\left(x\right)$
Apply the formula: $x\cdot x$$=x^2$, where $x=\sin\left(x\right)$
Apply the formula: $x\cdot x$$=x^2$, where $x=\cos\left(x\right)$
Apply the formula: $\sin\left(\theta \right)^2+\cos\left(\theta \right)^2$$=1$
II. Express the RHS in terms of sine and cosine and simplify
Start from the RHS (right-hand side)
Apply the trigonometric identity: $\sec\left(\theta \right)$$=\frac{1}{\cos\left(\theta \right)}$
Apply the trigonometric identity: $\csc\left(\theta \right)$$=\frac{1}{\sin\left(\theta \right)}$
Apply the formula: $\frac{a}{b}\frac{c}{f}$$=\frac{ac}{bf}$, where $a=1$, $b=\cos\left(x\right)$, $c=1$, $a/b=\frac{1}{\cos\left(x\right)}$, $f=\sin\left(x\right)$, $c/f=\frac{1}{\sin\left(x\right)}$ and $a/bc/f=\frac{1}{\cos\left(x\right)}\frac{1}{\sin\left(x\right)}$
III. Choose what side of the identity are we going to work on
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 $\frac{1}{\cos\left(x\right)\sin\left(x\right)}$
IV. Check if we arrived at the expression we wanted to prove
Since we have reached the expression of our goal, we have proven the identity
