Final Answer
Step-by-step Solution
Specify the solving method
Starting from the left-hand side (LHS) of the identity
Combine fractions with different denominator using the formula: $\displaystyle\frac{a}{b}+\frac{c}{d}=\frac{a\cdot d + b\cdot c}{b\cdot d}$
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: $(a+b)(a-b)=a^2-b^2$.
Factor the polynomial $\sin\left(x\right)\left(\sec\left(x\right)-1\right)+\sin\left(x\right)\left(\sec\left(x\right)+1\right)$ by it's greatest common factor (GCF): $\sin\left(x\right)$
Subtract the values $1$ and $-1$
Apply the trigonometric identity: $\sec\left(\theta \right)^2-1$$=\tan\left(\theta \right)^2$
Applying the trigonometric identity: $\sin\left(\theta \right)\sec\left(\theta \right) = \tan\left(\theta \right)$
Simplify the fraction by $\tan\left(x\right)$
Applying the trigonometric identity: $\cot\left(\theta\right)=\frac{1}{\tan\left(\theta\right)}$
Since we have reached the expression of our goal, we have proven the identity