Class CBSE Class 12 Mathematics Derivatives Q #1269
KNOWLEDGE BASED
UNDERSTAND
2 Marks 2024 AISSCE(Board Exam) VSA
If $y=cosec(cot^{-1}x)$, then prove that $\sqrt{1+x^{2}}\frac{dy}{dx}-x=0$ .
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Detailed Solution

Step 1: Express $cot^{-1}x$ as $\theta$

Let $\theta = cot^{-1}x$. This implies that $cot\theta = x$.

Step 2: Find $cosec\theta$

We know that $cosec^{2}\theta = 1 + cot^{2}\theta$. Therefore, $cosec^{2}\theta = 1 + x^{2}$, and $cosec\theta = \sqrt{1+x^{2}}$.

Step 3: Express $y$ in terms of $x$

Since $y = cosec(cot^{-1}x) = cosec(\theta)$, we have $y = \sqrt{1+x^{2}}$.

Step 4: Differentiate $y$ with respect to $x$

Differentiating $y = \sqrt{1+x^{2}}$ with respect to $x$, we get: $$ \frac{dy}{dx} = \frac{1}{2\sqrt{1+x^{2}}} \cdot 2x = \frac{x}{\sqrt{1+x^{2}}} $$

Step 5: Verify the given equation

Now, we need to prove that $\sqrt{1+x^{2}}\frac{dy}{dx} - x = 0$. Substituting the value of $\frac{dy}{dx}$, we have: $$ \sqrt{1+x^{2}} \cdot \frac{x}{\sqrt{1+x^{2}}} - x = x - x = 0 $$ Thus, the given equation is verified.

Final Answer: Verified: $\sqrt{1+x^{2}}\frac{dy}{dx}-x=0$

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Pedagogical Audit
Bloom's Analysis: This is an UNDERSTAND question because the student needs to understand the relationship between trigonometric functions and their inverses, and then apply differentiation rules to prove the given equation.
Knowledge Dimension: CONCEPTUAL
Justification: The question requires understanding the concepts of inverse trigonometric functions, trigonometric identities, and differentiation. It's not just recall of facts or application of a specific algorithm.
Syllabus Audit: In the context of CBSE Class 12, this is classified as KNOWLEDGE. The question directly tests the student's understanding of inverse trigonometric functions and differentiation, which are core concepts in the syllabus.

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