Final answer to the problem
Step-by-step Solution
Specify the solving method
Starting from the right-hand side (RHS) of the identity
Applying the trigonometric identity: $1+\tan\left(\theta \right)^2 = \sec\left(\theta \right)^2$
Applying the secant identity: $\displaystyle\sec\left(\theta\right)=\frac{1}{\cos\left(\theta\right)}$
Divide fractions $\frac{\tan\left(x\right)^2-1}{\frac{1}{\cos\left(x\right)^2}}$ with Keep, Change, Flip: $a\div \frac{b}{c}=\frac{a}{1}\div\frac{b}{c}=\frac{a}{1}\times\frac{c}{b}=\frac{a\cdot c}{b}$
Applying the pythagorean identity: $\cos^2(\theta)=1-\sin(\theta)^2$
Multiply the single term $1-\sin\left(x\right)^2$ by each term of the polynomial $\left(\tan\left(x\right)^2-1\right)$
Simplifying
Simplify $\tan\left(x\right)^2\cos\left(x\right)^2$ into by applying trigonometric identities
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