Prove and Solve: Trigonometric Identity Equation
Prove that \(1+\tan^2(x)=\sec^2(x)\) by dividing the identity \(\sin^2(x)+\cos^2(x)=1\) by \(\cos^2(x)\). Then, using this identity, solve the equation \(\sec^2(x)-2=\tan^2(x)\) for \(x\) in the interval \([0,2\pi)\).
Uniqueness of Solution via Contradiction
Prove that the equation \(x^2 + 1 = 2x\) has at most one real solution by assuming it has two distinct solutions and deriving a contradiction. Clearly justify each step of your reasoning.