If sinθ + cosecθ = 2 for an acute angle θ in trigonometry, then what is the value of sinⁿθ + cosecⁿθ for any positive integer n?

Introduction / Context:
This problem tests understanding of trigonometric functions and algebraic manipulation of expressions involving a function and its reciprocal. The given condition sinθ + cosecθ = 2 is very restrictive and leads to a special value of sinθ. Once that value is identified, it becomes easy to evaluate sinⁿθ + cosecⁿθ for any positive integer n.

Given Data / Assumptions:

• θ is an acute angle, so 0° < θ < 90° in usual trigonometric context.
• sinθ + cosecθ = 2.
• cosecθ is the reciprocal of sinθ, so cosecθ = 1 / sinθ.
• n is a positive integer, so n = 1, 2, 3 and so on.

Concept / Approach:
First we rewrite the given equation using the reciprocal relationship. Set x = sinθ, with 0 < x ≤ 1 for an acute or right angle. Then the condition becomes x + 1/x = 2. This is an algebraic equation in x. We solve this equation and observe that the only possible positive real solution is x = 1. Once x is known, all powers xⁿ and (1/x)ⁿ can be evaluated. The sum sinⁿθ + cosecⁿθ then becomes a simple constant independent of n.

Step-by-Step Solution:
Let x = sinθ, then cosecθ = 1/x.Given equation becomes x + 1/x = 2.Multiply both sides by x to clear the denominator: x^2 + 1 = 2x.Rearrange: x^2 - 2x + 1 = 0.Factor this quadratic: (x - 1)^2 = 0.Therefore x = 1, so sinθ = 1 and hence cosecθ = 1.Then sinⁿθ + cosecⁿθ = 1ⁿ + 1ⁿ = 1 + 1 = 2 for any positive integer n.

Verification / Alternative check:
Check whether any other positive value of x can satisfy x + 1/x = 2. Using the inequality x + 1/x ≥ 2 for all positive x, with equality only when x = 1, we see that x must be exactly 1. Thus the pair (sinθ, cosecθ) is fixed as (1, 1), and raising these values to any positive integer power leaves them equal to 1. Therefore the sum always remains 2, which matches our derived answer.

Why Other Options Are Wrong:
0 or 1 or 1/n would require different values for sinθ or cosecθ, which contradict the strict equality x + 1/x = 2. Option 4 would require each term to be 2, which is impossible because sinθ cannot exceed 1. Hence none of these alternative values are consistent with the given condition.

Common Pitfalls:
Some students may misinterpret the condition and think that θ can vary, but in fact the relation forces sinθ to a unique value. Others may incorrectly assume sinθ = 0, which is not allowed for cosecθ, or forget that 1/x is the reciprocal. Additionally, confusion may arise around the word acute, but the algebraic result still forces sinθ to 1 in the limiting right angle case commonly accepted in many exam questions.

Final Answer:
The value of sinⁿθ + cosecⁿθ for any positive integer n is 2.