Trigonometry & Inverse Trigonometry
Trigonometry
star_batch_jee_advanced_2025
Grade 11
Question:
Given that for $a, b, c, d \in \mathbb{R}$, if $a\sec(200°) - c\tan(200°) = d$ and $b\sec(200°) + d\tan(200°) = c$, then find the value of $\left(\frac{a^2 + b^2 + c^2 + d^2}{bd - ac}\right)\sin 20°$
Step-by-Step Solution
Key Concept: Add the two squared equations to establish a relationship, then manipulate to eliminate variables systematically.
We are given two equations with $a^2 \sec^2 200°$ and $b^2 \sec^2 200°$. Adding them yields $a^2 + b^2 = c^2 + d^2$. Further manipulation with the expressions $(a\sec 200° - c\tan 200°)^2 = d^2$ and $(b\sec 200° + d\tan 200°)^2 = c^2$ leads to $(c^2 + d^2)(2\tan^2 200°) = (2ac - 2bd)\sec 200°\tan 200°$, simplifying to $\frac{2(c^2+d^2)}{ac-bd} = \frac{-2}{\sin 20°}$.
Correct Answer: 2