Final answer to the problem
Step-by-step Solution
Specify the solving method
Starting from the left-hand side (LHS) of the identity
Multiply and divide the fraction $\frac{1+\cos\left(x\right)}{\sin\left(x\right)}$ by the conjugate of it's numerator $1+\cos\left(x\right)$
Multiplying fractions $\frac{1+\cos\left(x\right)}{\sin\left(x\right)} \times \frac{1-\cos\left(x\right)}{1-\cos\left(x\right)}$
The sum of two terms multiplied by their difference is equal to the square of the first term minus the square of the second term. In other words: $(a+b)(a-b)=a^2-b^2$.
Apply the trigonometric identity: $1-\cos\left(\theta \right)^2$$=\sin\left(\theta \right)^2$
Simplify the fraction $\frac{\sin\left(x\right)^2}{\sin\left(x\right)\left(1-\cos\left(x\right)\right)}$ by $\sin\left(x\right)$
Since we have reached the expression of our goal, we have proven the identity