Final Answer
Step-by-step Solution
Specify the solving method
Starting from the right-hand side (RHS) of the identity
Use the trigonometric identities: $\displaystyle\tan\left(\theta\right)=\frac{\sin\left(\theta\right)}{\cos\left(\theta\right)}$ and $\displaystyle\cot\left(\theta\right)=\frac{\cos\left(\theta\right)}{\sin\left(\theta\right)}$
Rewrite $\frac{\tan\left(x\right)^2-1}{\tan\left(x\right)^2+1}$ in terms of sine and cosine functions
Combine $\frac{\sin\left(x\right)^2}{\cos\left(x\right)^2}+1$ in a single fraction
Applying the pythagorean identity: $\sin^2\left(\theta\right)+\cos^2\left(\theta\right)=1$
Simplify the fraction $\frac{\sin\left(x\right)^2}{\cos\left(x\right)^2}+1$ into $\frac{1}{\cos\left(x\right)^2}$
Divide fractions $\frac{\frac{\sin\left(x\right)^2}{\cos\left(x\right)^2}-1}{\frac{1}{\cos\left(x\right)^2}}$ with Keep, Change, Flip: $a\div \frac{b}{c}=\frac{a}{1}\div\frac{b}{c}=\frac{a}{1}\times\frac{c}{b}=\frac{a\cdot c}{b}$
Combine all terms into a single fraction with $\cos\left(x\right)^2$ as common denominator
Multiplying the fraction by $\cos\left(x\right)^2$
Simplify the fraction $\frac{\left(\sin\left(x\right)^2-\cos\left(x\right)^2\right)\cos\left(x\right)^2}{\cos\left(x\right)^2}$ by $\cos\left(x\right)^2$
Multiplying the fraction by $\cos\left(x\right)^2$
Applying the trigonometric identity: $\sin\left(\theta \right)^2 = 1-\cos\left(\theta \right)^2$
Combining like terms $-\cos\left(x\right)^2$ and $-\cos\left(x\right)^2$
Since we have reached the expression of our goal, we have proven the identity