If (cos x)y = (sin y)x, find dy dx . - Vidyadip

Question

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

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Answer

(cos x)^y = (sin y)^x

$\Rightarrow$ y log cos x = x log sin y

$\therefore$ log (cos x)_^.$\frac{\text{dy}}{\text{dx}}$ - y ^_.tan x = log sin y + x cot y ^_.$\frac{\text{dy}}{\text{dx}}$

$\therefore$$\frac{\text{dy}}{\text{dx}}$ (log cos x - x cot y) = log sin y + y tan x

$\therefore$$\frac{\text{dy}}{\text{dx}}=\frac{\log\sin\text{y + y tan x}}{\text{log cos x - x}\cdot\text{cot y}}.$

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