Question

Find \(\frac{dy}{dx}\), if x=log⁡(tan⁡t) and y=log⁡(sin⁡t).

(a) 2 cos^2⁡t

(b) cos^2⁡2t

(c) cos^2t

(d) -cos^2t

This question was posed to me during an interview.

My enquiry is from Derivatives of Functions in Parametric Forms topic in division Continuity and Differentiability of Mathematics – Class 12

Answer 1

Right answer is (c) cos^2t

The explanation is: Given that, x=log⁡(tan⁡t) and y=log⁡(sin⁡t)

\(\frac{dx}{dt}\)=\(\frac{1}{tan⁡t}.sec^2⁡t=cot⁡t sec^2⁡t\)

\(\frac{dy}{dt}=\frac{1}{sin⁡ \,t}.cos⁡ \,t=cot⁡ \,t\)

∴\(\frac{dy}{dx}\)=\(\frac{dy}{dt}.\frac{dt}{dx}=\frac{cot⁡\,t}{cot\,⁡t sec^2⁡t}=\frac{1}{sec^2⁡t}=cos^2⁡t\).