Final Answer
Step-by-step Solution
Specify the solving method
I. Express the LHS in terms of sine and cosine and simplify
Start from the LHS (left-hand side)
Applying the pythagorean identity: $\cos^2(\theta)=1-\sin(\theta)^2$
Multiply the single term $-2$ by each term of the polynomial $\left(1-\sin\left(x\right)^2\right)$
Add the values $1$ and $-2$
II. Express the RHS in terms of sine and cosine and simplify
Start from the RHS (right-hand side)
Rewrite $\tan\left(x\right)^2$ in terms of sine and cosine
Rewrite $\tan\left(x\right)^2$ in terms of sine and cosine
Simplify the fraction $\frac{\sin\left(x\right)^2}{\cos\left(x\right)^2}+1$ into $\frac{1}{\cos\left(x\right)^2}$
Combine all terms into a single fraction with $\cos\left(x\right)^2$ as common denominator
Simplify the fraction $\frac{\frac{\sin\left(x\right)^2-\cos\left(x\right)^2}{\cos\left(x\right)^2}}{\frac{1}{\cos\left(x\right)^2}}$
III. Choose what side of the identity are we going to work on
To prove an identity, we usually begin to work on the side of the equality that seems to be more complicated, or the side that is not expressed in terms of sine and cosine. In this problem, we will choose to work on the right side $\sin\left(x\right)^2-\cos\left(x\right)^2$ to reach the left side $-1+2\sin\left(x\right)^2$
Applying the pythagorean identity: $\cos^2(\theta)=1-\sin(\theta)^2$
Simplify the product $-(1-\sin\left(x\right)^2)$
Multiply $-1$ times $-1$
Combining like terms $\sin\left(x\right)^2$ and $\sin\left(x\right)^2$
IV. Check if we arrived at the expression we wanted to prove
Since we have reached the expression of our goal, we have proven the identity