Trigonometric Ratios and Identities
Tangent and Cotangent
JEE Main 2023
Grade 11
Question:
If $\tan 15^\circ + \frac{1}{\tan 75^\circ} + \frac{1}{\tan 105^\circ} + \tan 195^\circ = 2a$, then the value of $\left(a + \frac{1}{a}\right)$ is:
(1) $4$
(2) $2\sqrt{3}$
(3) $2 + \sqrt{3}$
(4) $2 - \sqrt{3}$
Step-by-Step Solution
Key Concept: Simplify each term: $1/\tan 75^\circ = \tan 15^\circ$, $\tan 105^\circ = -\tan 75^\circ$ so $1/\tan 105^\circ = -\tan 15^\circ$, and $\tan 195^\circ = \tan 15^\circ$. Add all four, find $a$, then compute $a + 1/a$.
The detailed step-by-step mathematical proof is available inside the Mathbee app workspace.
Correct Answer: (1)