Which of the following equation has no solution (a) \(4 \sin \theta+3 \cos \theta=1\) (b) \(\operatorname{cosec} \theta \cdot \sec \theta=1\) (c) \(\sin \theta \cdot \cos \theta=(1 / 2)\) (d) \(\operatorname{cosec} \theta-\sec \theta=\operatorname{cosec} \theta \cdot \sec \theta\)

To analyze the equation \(4\sin\theta + 3\cos\theta = 1\), we can use the following trigonometric inequality: \(\lvert\sin\theta\rvert \le 1\) and \(\lvert\cos\theta\rvert \le 1\). Now, we can rewrite the equation as a weighted sum of sine and cosine terms, which will be in the range of their respective limits. The lowest value will be when \(\sin\theta=-1\) and \(\cos\theta=-1\), so the expression becomes \(4(-1)+3(-1)\), which equals -7. The highest value will be when \(\sin\theta=1\) and \(\cos\theta=1\), so the expression becomes \(4(1)+3(1)\), which equals 7. Since the range of possible values for the given expression is between -7 and 7, the equation is true for some values of θ. Hence, this equation has a solution.

To analyze the equation \(\cosec\theta \cdot \sec\theta = 1\), we can rewrite it using the reciprocal trigonometric identities, \(\cosec\theta = \frac{1}{\sin\theta}\) and \(\sec\theta = \frac{1}{\cos\theta}\). The equation then becomes \(\frac{1}{\sin\theta} \cdot \frac{1}{\cos\theta} = 1\), or \(\frac{1}{\sin\theta\cos\theta} = 1\). Multiplying both sides by \(\sin\theta\cos\theta\), we have \(1 = \sin\theta\cos\theta\). Since the range of \(\sin\theta\) and \(\cos\theta\) is [-1, 1], the product \(\sin\theta\cos\theta\) will also be in the range [0, 1], so there will be some values of θ for which the equation is true. Hence, this equation has a solution.

The equation is \(\sin\theta\cdot\cos\theta = \frac{1}{2}\). The maximum value of \(\sin\theta\) and \(\cos\theta\) is 1. The maximum value of the product \(\sin\theta\cdot\cos\theta\) is therefore also 1, and it occurs when \(θ = \frac{\pi}{4}\). Since the range of the product is between 0 and 1, the equation is true for some values of θ. Hence, this equation has a solution.

The given equation is \(\cosec\theta-\sec\theta=\cosec\theta\cdot\sec\theta\). By adding \(\sec\theta\) to both sides, we get \(\cosec\theta=\cosec\theta\cdot\sec\theta+\sec\theta\). Now, divide both sides by \(\cosec\theta\): \(1 = \sec\theta + \frac{\sec\theta}{\cosec\theta}\) Setting \(\sec\theta = A = \frac{1}{\cos\theta}\) and \(\cosec\theta = B = \frac{1}{\sin\theta}\), the equation becomes \(1 = A + \frac{A}{B}\) Multiplying by B, we get \(B = A\cdot B + A\) By rearranging, we get \(A\cdot B = B - A\) Since \(B = \frac{1}{\sin\theta}\), we can rewrite the equation in terms of \(\sin\theta\): \(\frac{1}{\cos\theta} \cdot \frac{1}{\sin\theta} = \frac{1}{\sin\theta} - \frac{1}{\cos\theta}\) Since \(\sin(\pi - \theta) = \sin\theta\) and \(\cos(\pi - \theta) = -\cos\theta\), the equation can be written as \(\tan(\pi - \theta) = \cot(\theta)\) This is a contradiction, as it states that the tangent of an angle is equal to the cotangent of the same angle. Hence, this equation has no solution. The correct answer is (d).