Final Answer
true
Step-by-step Solution
Specify the solving method
1
Starting from the right-hand side (RHS) of the identity
$\frac{\sin\left(x\right)}{1-\cos\left(x\right)}$
2
Multiply and divide the fraction $\frac{\sin\left(x\right)}{1-\cos\left(x\right)}$ by the conjugate of it's denominator $1-\cos\left(x\right)$
$\frac{\sin\left(x\right)}{1-\cos\left(x\right)}\frac{1+\cos\left(x\right)}{1+\cos\left(x\right)}$
3
Multiplying fractions $\frac{\sin\left(x\right)}{1-\cos\left(x\right)} \times \frac{1+\cos\left(x\right)}{1+\cos\left(x\right)}$
$\frac{\sin\left(x\right)\left(1+\cos\left(x\right)\right)}{\left(1-\cos\left(x\right)\right)\left(1+\cos\left(x\right)\right)}$
4
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: $(a+b)(a-b)=a^2-b^2$.
$\frac{\sin\left(x\right)\left(1+\cos\left(x\right)\right)}{1-\cos\left(x\right)^2}$
5
Apply the trigonometric identity: $1-\cos\left(\theta \right)^2$$=\sin\left(\theta \right)^2$
$\frac{\sin\left(x\right)\left(1+\cos\left(x\right)\right)}{\sin\left(x\right)^2}$
Why is 1 - cos(x)^2 = sin(x)^2 ?
6
Simplify the fraction by $\sin\left(x\right)$
$\frac{1+\cos\left(x\right)}{\sin\left(x\right)}$
7
Since we have reached the expression of our goal, we have proven the identity
true
Final Answer
true