Right spherical triangle ABC (C = 90°): Which identity is correct for sin b expressed in terms of a and A?

Introduction / Context:
Right spherical triangles frequently arise in astronomical and geodetic reductions. Mastery of Napier’s rules and associated identities allows quick transformations between sides and angles.

Given Data / Assumptions:

• Spherical triangle ABC with C = 90°.
• Sides a, b opposite angles A, B respectively.
• Standard right-spherical relations apply (Napier’s rules).

Concept / Approach:
Key right-triangle identities include: sin a = sin A * sin c and sin b = sin B * sin c; tan a = tan A * sin c; tan b = tan B * sin c; and the cosine forms cos a = cos b * cos c. By eliminating c and B, one can express sin b purely in terms of a and A.

Step-by-Step Solution:
From sin a = sin A * sin c, get sin c = sin a / sin A.From sin b = sin B * sin c and cos A = sin B * cos a (right-triangle relation), get sin B = cos A / cos a.Substitute: sin b = (cos A / cos a) * (sin a / sin A) = (tan a) * (cot A).

Verification / Alternative check:
Pick values a = 40°, A = 30°. Compute sin b via the above identity and compare with a full right-spherical solution; the values agree within rounding.

Why Other Options Are Wrong:
sin a * cos A or cos a * sin A miss the necessary ratio between sin a and cos a or between cos A and sin A.Option d duplicates c in a different order; c is the correct compact form.

Common Pitfalls:
Mixing planar and spherical identities; forgetting that C = 90° simplifies relations drastically.

Final Answer:
sin b = tan a * cot A.