Trigonometric Ratios and Identities
Infinite Series
JEE Main 2020
Grade 11
Question:
Let $x = \sum_{n=0}^\infty (-1)^n \tan^{2n} \theta$ and $y = \sum_{n=0}^\infty \cos^{2n} \theta$, where $\theta \in (0, \pi/4)$. Then which holds?
(1) $x + y - xy = 1$
(2) $xy = y - 1$
(3) $x + y + xy = 1$
(4) $y(1 + x) = 1$
Step-by-Step Solution
Key Concept: Sum $x$ as a geometric series with ratio $-\tan^2 \theta$; sum $y$ with ratio $\cos^2 \theta$. Express $x$ and $y$ in terms of $\sin \theta$ and $\cos \theta$, then verify each option.
The detailed step-by-step mathematical proof is available inside the Mathbee app workspace.
Correct Answer: (2)