Final answer to the problem
Step-by-step Solution
How should I solve this problem?
- Prove from LHS (left-hand side)
- Prove from RHS (right-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
- Load more...
Starting from the left-hand side (LHS) of the identity
Applying the tangent identity: $\displaystyle\tan\left(\theta\right)=\frac{\sin\left(\theta\right)}{\cos\left(\theta\right)}$
Applying the trigonometric identity: $\cot\left(\theta \right) = \frac{\cos\left(\theta \right)}{\sin\left(\theta \right)}$
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
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
Rewrite the sum of fractions as a single fraction with the same denominator
When multiplying two powers that have the same base ($\sin\left(x\right)$), you can add the exponents
When multiplying two powers that have the same base ($\cos\left(x\right)$), you can add the exponents
Applying the pythagorean identity: $\sin^2\left(\theta\right)+\cos^2\left(\theta\right)=1$
Combine and simplify all terms in the same fraction with common denominator $\cos\left(x\right)\sin\left(x\right)$
Applying the trigonometric identity: $\displaystyle\sec\left(\theta\right)=\frac{1}{\cos\left(\theta\right)}$
The reciprocal sine function is cosecant: $\frac{1}{\sin(x)}=\csc(x)$
Since we have reached the expression of our goal, we have proven the identity
