Question on partial derivative

Question

If $u= \log(\tan x + \tan y)$, prove that $$\sin(2x) \, \frac{du}{dx} + \sin(2y) \, \frac{du}{dy} = 2$$.

Answer 1

For $u = \ln(\tan x + \tan y)$ the derivatives with respect to $x$ and $y$ are \begin{align} \frac{du}{dx} &= \frac{sec^{2}(x)}{\tan x + \tan y} \\ \frac{du}{dy} &= \frac{sec^{2}(y)}{\tan x + \tan y}. \end{align} Now, \begin{align} \sin(2x) \, \frac{du}{dx} + \sin(2y) \, \frac{du}{dy} &= 2 \, \frac{\tan x + \tan y}{\tan x + \tan y} = 2. \end{align}