Question

Match List I (Trigonometric Square Root Expressions) with List II (Simplified Functions):

List I		List II
A.	\(\sqrt{1 - \text{Kotijya}^2}\)	I.	Kotijya
B.	\(\sqrt{1 - \text{Jya}^2}\)	II.	Jya
C.	\(\text{Sparsha-rekha}\) (Tangent)	III.	\(\frac{\text{Kotijya}}{\text{Jya}}\)
D.	\(\text{Koti-sparsha-rekha}\) (Cotangent)	IV.	\(\frac{\text{Jya}}{\text{Kotijya}}\)

• A. A-I, B-II, C-III, D-IV
• B. A-III, B-IV, C-II, D-I
• C. A-IV, B-III, C-I, D-II
• D. A-II, B-I, C-IV, D-III

Hint:

Remember: Jya = Sine, Kotijya = Cosine. Sparsha (Touch) = Tangent. The term "Koti" usually refers to the "complementary" or "adjacent" side.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
Indian trigonometry is fundamentally based on the Pythagorean relationship within a unit circle (though scaled by Trijya). The Sine is 'Jya' and the Cosine is 'Kotijya'. The derived ratios for the tangent and cotangent were defined by the length of shadow segments, known as 'Sparsha-rekhas'.

Step 2: Detailed Matching using Pythagorean Identities:
1. Identity for Jya (A): Since \(\sin^2 \theta + \cos^2 \theta = 1\), we have \(\sin \theta = \sqrt{1 - \cos^2 \theta}\). In Siddhantic terms:
\[ \text{Jya} = \sqrt{R^2 - \text{Kotijya}^2} \]
(Assuming $R=1$ for simplicity in matching). Therefore, A matches with II.
2. Identity for Kotijya (B): Similarly, \(\cos \theta = \sqrt{1 - \sin^2 \theta}\). In Siddhantic terms:
\[ \text{Kotijya} = \sqrt{R^2 - \text{Jya}^2} \]
Therefore, B matches with I.
3. Tangent (C): 'Sparsha-rekha' is the Sanskrit term for Tangent. In terms of primary functions:
\[ \tan \theta = \frac{\sin \theta}{\cos \theta} \rightarrow \text{Sparsharekha} = \frac{\text{Jya}}{\text{Kotijya}} \]
Therefore, C matches with IV.
4. Cotangent (D): 'Koti-sparsharekha' is the Cotangent.
\[ \cot \theta = \frac{\cos \theta}{\sin \theta} \rightarrow \text{Koti-sparsharekha} = \frac{\text{Kotijya}}{\text{Jya}} \]
Therefore, D matches with III.

Step 3: Verification:
Sequence: A-II, B-I, C-IV, D-III. This matches Option (4).

Step 4: Final Answer:
The list correctly identifies the fundamental Pythagorean and reciprocal relationships in Jya-ganita.