Prove that the function f given by `f (x) = log cos x`is strictly decreasing on `(0,pi/2)`and strictly increasing on `(pi/2,pi)`prove that the function f given by ` f(x) = log sin x ` is strictly decreasing on `( 0 , pi/2)` and strictly increasing on `( pi/2 , pi) `..

Explanation

To prove the properties of the functions \( f(x) = \log(\cos x) \) and \( g(x) = \log(\sin x) \) on the specified intervals, we will find the derivatives of these functions and analyze their signs.

### Step 1: Analyze \( f(x) = \log(\cos x) \)

1. **Find the derivative**:
\[ f'(x) = \frac{d}{dx}[\log(\cos x)] = \frac{1}{\cos x} \cdot (-\sin x) = -\tan x \]
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