The value of the expression sin x + cosec x = 2, then the value of sin⁷ x + cosec⁷ x is:

Correct — C. Rule: t + 1⁄t = 2 forces t = 1, because it rearranges to (t − 1)² = 0.

Put t = sin x, so cosec x = 1⁄t:
t + 1⁄t = 2 → t² − 2t + 1 = 0
(t − 1)² = 0 → t = 1

So sin x = 1 and cosec x = 1, which happens at x = 90°.

sin⁷x + cosec⁷x = 1⁷ + 1⁷ = 2 → option (c)

Why the others are wrong

• (a)
  1 — 1 is sin⁷x by itself. Once sin x = 1 its reciprocal is also 1, and the expression asks for both terms, so the total is two.
• (b)
  4 — 4 is the square of the given sum. Squaring actually yields sin²x + cosec²x = 4 − 2 = 2, so even that route does not reach 4.
• (d)
  0 — 0 would need the two terms to cancel, but sin x and cosec x are reciprocals and always share a sign, so their sum is never zero for any real x.

This is not really a trigonometry question. It is the algebraic fact that a positive number and its reciprocal add to at least 2, with equality only when the number is 1.

Once sin x is pinned to 1, every power behaves the same way: if t + 1⁄t = 2 then tⁿ + 1⁄tⁿ = 2 for every n, so the exponent 7 is decoration.

The mirror case is worth carrying too: t + 1⁄t = −2 forces t = −1, and then the odd powers give −2.

Key facts

• For real t > 0, t + 1⁄t ≥ 2, and equality holds exactly at t = 1.
• sin x = 1 at x = 90°, where cosec x is also 1.
• If t + 1⁄t = 2 then tⁿ + 1⁄tⁿ = 2 for every positive integer n.

Common traps

• Raising the given sum to the seventh power instead of the variable.
• Assuming a high exponent must produce a large answer.
• Forgetting that sin x = 1 is the single admissible value here.

No directly related past PYQ was found.