Qi Express as a simple \( \log , \ln \sin x-\ln \cos x \)

To simplify \( \log(\sin x) - \log(\cos x) \), we can use the properties of logarithms. First, the subtraction of logs turns into a division: \[ \log(\sin x) - \log(\cos x) = \log\left(\frac{\sin x}{\cos x}\right) \] Now, recognizing that \( \frac{\sin x}{\cos x} = \tan x \), we can rewrite the expression as: \[ \log(\sin x) - \log(\cos x) = \log(\tan x) \] So, our final simplified version is: \[ \log(\tan x) \] For the natural logarithm, we similarly have: \[ \ln(\sin x) - \ln(\cos x) = \ln\left(\frac{\sin x}{\cos x}\right) = \ln(\tan x) \] Thus, whether you're working in base \( e \) or base 10, this expression beautifully simplifies to: \(\log(\tan x)\) or \(\ln(\tan x)\)!