(i) Given that \( y=3 \sin ^{2} x+\cos x \), show that \( y+\cot x \frac{\mathrm{~d} y}{\mathrm{~d} x}=k\left(1+\cos ^{2} x\right) \), where \( k \) is an integer.

To show that \[ y + \cot x \frac{dy}{dx} = k(1 + \cos^2 x), \] where \( y = 3 \sin^2 x + \cos x \), we first need to compute \( \frac{dy}{dx} \). ### Step 1: Compute \( \frac{dy}{dx} \) Using the chain rule and the derivatives of sine and cosine, we have: \[ \frac{dy}{dx} = \frac{d}{dx}(3 \sin^2 x) + \frac{d}{dx}(\cos x). \] Calculating each term: 1. For \( 3 \sin^2 x \): \[ \frac{d}{dx}(3 \sin^2 x) = 3 \cdot 2 \sin x \cos x = 6 \sin x \cos x. \] 2. For \( \cos x \): \[ \frac{d}{dx}(\cos x) = -\sin x. \] Combining these results, we get: \[ \frac{dy}{dx} = 6 \sin x \cos x - \sin x = \sin x (6 \cos x - 1). \] ### Step 2: Substitute \( \frac{dy}{dx} \) into the equation Now we substitute \( y \) and \( \frac{dy}{dx} \) into the left-hand side of the equation: \[ y + \cot x \frac{dy}{dx} = (3 \sin^2 x + \cos x) + \cot x \cdot \sin x (6 \cos x - 1). \] Recall that \( \cot x = \frac{\cos x}{\sin x} \), so we can rewrite the second term: \[ \cot x \cdot \sin x (6 \cos x - 1) = \cos x (6 \cos x - 1) = 6 \cos^2 x - \cos x. \] Now, substituting this back into our equation gives: \[ y + \cot x \frac{dy}{dx} = (3 \sin^2 x + \cos x) + (6 \cos^2 x - \cos x). \] ### Step 3: Simplify the expression Combining the terms, we have: \[ y + \cot x \frac{dy}{dx} = 3 \sin^2 x + \cos x + 6 \cos^2 x - \cos x. \] The \( \cos x \) terms cancel out: \[ y + \cot x \frac{dy}{dx} = 3 \sin^2 x + 6 \cos^2 x. \] Using the identity \( \sin^2 x + \cos^2 x = 1 \), we can express \( \sin^2 x \) as \( 1 - \cos^2 x \): \[ 3 \sin^2 x = 3(1 - \cos^2 x) = 3 - 3 \cos^2 x. \] Thus, we can rewrite the expression: \[ y + \cot x \frac{dy}{dx} = (3 - 3 \cos^2 x) + 6 \cos^2 x = 3 + 3 \cos^2 x. \] ### Step 4: Factor out the common term Factoring out the common term gives: \[ y + \cot x \frac{dy}{dx} = 3(1 + \cos^2 x). \] ### Conclusion We can see that this matches the form \( k(1 + \cos^2 x) \) where \( k = 3 \). Therefore, we conclude that: \[ y + \cot x \frac{dy}{dx} = 3(1 + \cos^2 x), \] and thus \( k = 3 \). The final result is: \[ \boxed{3}. \]