Final Answer
Step-by-step Solution
Specify the solving method
Starting from the left-hand side (LHS) of the identity
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
Simplify the numerators
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
Applying the pythagorean identity: $\sin^2\left(\theta\right)+\cos^2\left(\theta\right)=1$
Add the values $1$ and $1$
Combine and simplify all terms in the same fraction with common denominator $\left(1+\cos\left(x\right)\right)\sin\left(x\right)$
Factor the polynomial $2+2\cos\left(x\right)$ by it's greatest common factor (GCF): $2$
Simplify the fraction $\frac{2\left(1+\cos\left(x\right)\right)}{\left(1+\cos\left(x\right)\right)\sin\left(x\right)}$ by $1+\cos\left(x\right)$
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