If $\tan A+\cot A=2$, find the value of ${\tan }^{2}A+{\cot }^{2}A$.

Square the given equation: $(\tan A+\cot A{)}^2={\tan }^{2}A+{\cot }^{2}A+2\tan A\cot A$. Since $\tan A\cot A=1$, this reads $4={\tan }^{2}A+{\cot }^{2}A+2$, so ${\tan }^{2}A+{\cot }^{2}A=2$. Answering 4 is the common slip of squaring each term separately and forgetting to subtract the cross term.