UPSC CDS 2 2019 Math Paper

Concept:Use the logarithmic product rule and trigonometric reciprocal identities to simplify the sum to $\log_{10}(1)$.Explanation:First, combine all logarithms using the product rule:$\log_{10}(\cos\theta) + \log_{10}(\sin\theta) + \log_{10}(\tan\theta) + \log_{10}(\cot\theta) + \log_{10}(\sec\theta) + \log_{10}(\csc\theta) = \log_{10}(\cos\theta \cdot \sin\theta \cdot \tan\theta \cdot \cot\theta \cdot \sec\theta \cdot \csc\theta)$Now replace each function with its equivalent ratio:$\tan\theta = \frac{\sin\theta}{\cos\theta}$, $\cot\theta = \frac{\cos\theta}{\sin\theta}$, $\sec\theta = \frac{1}{\cos\theta}$, $\csc\theta = \frac{1}{\sin\theta}$Multiply all six factors:$\cos\theta \cdot \sin\theta \cdot \frac{\sin\theta}{\cos\theta} \cdot \frac{\cos\theta}{\sin\theta} \cdot \frac{1}{\cos\theta} \cdot \frac{1}{\sin\theta}$Simplify step by step:$\cos\theta$ cancels with $\frac{1}{\cos\theta}$; $\sin\theta$ cancels with $\frac{1}{\sin\theta}$; and $\frac{\sin\theta}{\cos\theta} \cdot \frac{\cos\theta}{\sin\theta} = 1$Thus the entire product equals $1$.Hence, $\log_{10}(1) = 0$.Answer:B. 0