Napier’s Rules (spherical right triangle) — relation for the middle part According to Napier’s Rules of circular parts for a right-angled spherical triangle, the sine of the middle part equals the product of which functions of the two adjacent parts?

Introduction / Context:
Napier’s Rules of circular parts provide compact mnemonic relations for right-angled spherical triangles, widely used in astronomical surveying and navigation. They relate the five “circular parts” (other than the right angle) via trigonometric functions to solve unknown sides/angles efficiently.

Given Data / Assumptions:

• We consider a spherical right triangle with one angle = 90°.
• Napier’s circular parts exclude the right angle and include the remaining sides and complements of certain angles.
• Standard Napier formulation applies.

Concept / Approach:
Napier’s Rules state, among others: “The sine of any part is equal to the product of the tangents of the adjacent parts,” and also “The sine of any part is equal to the product of the cosines of the opposite parts.” The question explicitly asks for the relation using adjacent parts; therefore, the correct choice is the product of tangents of the two adjacent parts.

Step-by-Step Solution:

Verification / Alternative check:
The complementary rule using opposite parts reads: sin(middle) = cos(opposite 1) * cos(opposite 2), which is consistent but involves opposite, not adjacent, parts. Since the option list separates these cases, we must pick the “adjacent” version.

Why Other Options Are Wrong:

• (b) Sines of adjacent parts is not a Napier identity.
• (c) Cosines of adjacent parts corresponds to a different relation; Napier pairs cosines with opposite parts.
• (d) Cosines of opposite parts is a valid but different rule; the prompt specifically asks for adjacent parts.
• (e) Both (a) and (b) cannot be true together.

Common Pitfalls:
Mixing up “adjacent” versus “opposite” parts on Napier’s circle; forgetting that angles may appear as complements among the circular parts.

Final Answer:
Tangents of the two adjacent parts