If sin P + cosec P = 2 for an angle P (in degrees), what is the value of sin 7P + cosec 7P?

Introduction / Context:
This question explores the relationship between a trigonometric function and its reciprocal. You are given that sin P + cosec P = 2 and asked to find sin 7P + cosec 7P. The key idea is to use an inequality for sin P + 1/sin P and identify when equality occurs, which pins down the value of sin P and hence P itself.

Given Data / Assumptions:

• sin P + cosec P = 2.
• cosec P is defined, so sin P ≠ 0.
• Angles are measured in degrees.

Concept / Approach:
For any nonzero real number s, the expression s + 1/s satisfies s + 1/s ≥ 2 if s is positive, with equality only when s = 1. Here s = sin P. Thus the equation sin P + 1/sin P = 2 implies sin P = 1 and P is a special angle. Once we know P, we can compute 7P and then find sin 7P and cosec 7P directly from the unit circle.

Step-by-Step Solution:
Let s = sin P. Given s + 1/s = 2. Multiply both sides by s: s^2 + 1 = 2s. Rearrange: s^2 − 2s + 1 = 0. This factorises as (s − 1)^2 = 0. Thus s = 1, so sin P = 1. In degrees, sin P = 1 when P = 90° + 360°k, where k is any integer. Take P = 90° for simplicity. Then 7P = 7 × 90° = 630°. Reduce 630° modulo 360°: 630° − 360° = 270°. So sin 7P = sin 270° = −1. Then cosec 7P = 1/sin 7P = 1/(−1) = −1. Therefore sin 7P + cosec 7P = −1 + (−1) = −2.
Verification / Alternative check:
Using the general solution P = 90° + 360°k, 7P = 630° + 2520°k. Subtract multiples of 360° to get 270° + 360°k, for any integer k. The sine of angles of the form 270° + 360°k is always −1, so sin 7P + cosec 7P is always −2, independent of k.

Why Other Options Are Wrong:
Values 2, 3, 1 and 0 would require sin 7P or its reciprocal to be positive or zero, which contradicts the fact that sin 7P is always −1 for the set of permissible P. Option 2 corresponds to sin 7P = 1, which would only happen if 7P were of the form 90° + 360°k, not satisfied here.

Common Pitfalls:
A common error is to forget that sin P + 1/sin P ≥ 2 only for positive sin P, or to mis-handle the reduction of 630° to a standard angle. Another mistake is to assume P is 30° or 60° by guesswork, which does not satisfy the original equation. Always solve the algebraic condition on sin P first.

Final Answer:
The value of sin 7P + cosec 7P is −2.