WBJEE 2024 Math Paper

To determine whether the expression $\cos^2 \phi + \cos^2(\theta+\phi) - 2 \cos \theta \cos \phi \cos(\theta+\phi)$ depends on $\theta$ and $\phi$, we need to simplify it and check for any dependencies on these variables.Let's rewrite the expression:$\cos^2 \phi + \cos^2(\theta+\phi) - 2 \cos \theta \cos \phi \cos(\theta+\phi)$Consider the trigonometric identity for cosine of a sum:$\cos(\theta+\phi) = \cos \theta \cos \phi - \sin \theta \sin \phi$ However, using this identity directly seems cumbersome. Instead, let's look at another potential approach, focusing on the structure of the expression. Notice that it is a combination of squared cosines and a product of cosines. If we consider transforming the expression into terms of squares and products, we might be able to discern its dependencies more easily.Rewriting the expression in another form to recognize a potential independence can help. Let us test whether it can be simplified to a more recognizable form. After considerable algebraic manipulation, you would find it simplifies to: $\cos^2 \phi + \cos^2(\theta+\phi) - 2 \cos \theta \cos \phi \cos(\theta+\phi) = \sin^2 \theta$Thus, the given expression simplifies to:$\sin^2 \theta$From this result, it's clear that the simplified expression only depends on $\theta$.Hence, the correct option is:Option B: independent of $\phi$