Question

If $(\cos x{)}^y=(\cos y{)}^x$, find $\frac{dy}{dx}.$

Answer 1

$(\cos x{)}^y=(\cos y{)}^x$

Taking log on both sides, $y\ln \cos x=x\ln \cos y$

Differentiating both sides w.r.t $x$, we get

$\frac{dy}{dx}\times \ln \cos x+y\times \frac{-\sin x}{\cos x}=\ln \cos y+x\times \frac{-\sin y}{\cos y}\times \frac{dy}{dx}$

$\therefore \frac{dy}{dx}=\frac{\ln \cos y+y\tan x}{\ln \cos x+x\tan y}$