NCERT Class 12 · Ch 5 Miscellaneous Exercise · Q16
NCERT Class 12 Maths · Chapter 5 Continuity And Differentiability · Miscellaneous Exercise Question 16
If cos y=x cos(a+y), cos a≠±1, prove y′=cos²(a+y)/sin a.
💡 Concept Hints
- Identify the domain and the inner and outer functions, or the relevant one-sided limits.
- Apply the appropriate differentiation or limit rule, keep all chain factors, and check exceptional points.
✅ Step-by-Step Solution
Final Answer: y′=cos²(a+y)/sin a.
- Differentiate: −sin y·y′=cos(a+y)−x sin(a+y)y′.
- Thus y′=cos(a+y)/[x sin(a+y)−sin y]. The original equation has cos(a+y)≠0, since otherwise cos y=0 would force sin a=0.
- Substitute x=cos y/cos(a+y). The denominator becomes [cos y sin(a+y)−sin y cos(a+y)]/cos(a+y)=sin a/cos(a+y).
- The condition on cos a gives sin a≠0. Division yields the claimed formula.
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