Quadratic Trigonometric Differential Identities continues the higher-order differentiation sequence. All formulae are used on intervals where the displayed expressions are defined. Let $p^2=a^2\cos^2\theta+b^2\sin^2\theta$, where $a,b$ are constants and $p>0$ on the interval considered. :::theorem[Differential Identity] If $p^2=a^2\cos^2\theta+b^2\sin^2\theta$, then $$\begin{equation*} p\frac{d^2p}{d\theta^2}+\left(\frac{dp}{d\theta}\right)^2+p^2=a^2+b^2. \end{equation*}$$ ::: :::proof Differentiating twice and simplifying gives $$\begin{equation*} p\frac{d^2p}{d\theta^2}+\left(\frac{dp}{d\theta}\right)^2=(b^2-a^2)(\cos^2\theta-\sin^2\theta). \end{equation*}$$ Adding $p^2$ and substituting the original expression for $p^2$ gives $a^2+b^2$. Hence the identity follows. $\square$ ::: :::exercise[Practice Questions] 1. State the main result of this lesson with its hypotheses. 2. Apply the method to a simple example. 3. Identify one restriction that must be checked. ::: :::answer[Answers] 1. The result must include the differentiability and nonzero-denominator conditions. 2. Work term by term and verify a low-order case. 3. The restriction comes from the original domain or from a division made in the derivation. ::: :::faq[Frequently Asked Questions] Q: Why are hypotheses stated before the formula? A: They identify the interval, differentiability order, and excluded values needed for a valid calculation. Q: What is the common mistake? A: The common mistake is to use the displayed formula after losing a domain or nonzero-denominator condition. Q: How can I check my result? A: Substitute a small value of $n$ or differentiate once directly and compare with the formula. ::: :::call-to-action[Continue Learning] subtitle: Review the result, complete the practice questions, and continue through the higher-order derivatives sequence. button: Next Lesson|/real-analysis/functions-of-a-single-variable-ii/higher-order-derivatives/partial-fractions-in-repeated-differentiation button-ghost: Review Higher Order Derivatives|/real-analysis/functions-of-a-single-variable-ii/higher-order-derivatives :::