These go beyond the textbook exercise — general/abstract proofs and reverse-engineering problems using fresh numbers, at HOTS level for board exam extension questions.
Q1 (General proof). Prove that (sinθ+cosθ)² + (sinθ−cosθ)² = 2, for any angle θ.
Solution: Expanding both squares: (sin²θ+2sinθcosθ+cos²θ) + (sin²θ−2sinθcosθ+cos²θ) = 2sin²θ+2cos²θ = 2(sin²θ+cos²θ) = 2. Proved for all θ, not just a specific value.
Q2 (General proof, harder). Prove that sec A (1 − sin A)(sec A + tan A) = 1.
Solution: (1−sinA)(secA+tanA) = secA+tanA−sinA·secA−sinA·tanA. Since sinA·secA = tanA, this becomes secA−sinA·tanA = secA−sin²A/cosA = (1−sin²A)/cosA = cos²A/cosA = cosA. So the full expression = secA·cosA = 1.
Q3 (General proof, 3D-style). If x = r sinA cosB, y = r sinA sinB, z = r cosA, prove x²+y²+z² = r².
Solution: x²+y² = r²sin²A(cos²B+sin²B) = r²sin²A. Adding z²: r²sin²A + r²cos²A = r².
Q4 (Reverse-engineering). If tanθ+cotθ = 2, find tan²θ+cot²θ without assuming θ = 45°.
Solution: Square both sides: (tanθ+cotθ)² = tan²θ+2+cot²θ = 4. So tan²θ+cot²θ = 4−2 = 2.
Q5 (Fresh-numbers quadratic-in-trig). Find x (0°≤x≤90°) if 2sin²x + 3cos x = 3.
Solution: Substitute sin²x = 1−cos²x: 2−2cos²x+3cosx = 3 ⇒ 2cos²x−3cosx+1 = 0. Solving the quadratic: cosx = (3±1)/4 = 1 or 1/2. cos x = 1/2 gives the meaningful acute-angle solution, x = 60°.
Q6 (Assertion-Reason). Assertion (A): If cosθ = 12/13, then sinθ = 5/13.
Reason (R): For any angle, sin²θ+cos²θ = 1.
Solution: sinθ = √(1−144/169) = √(25/169) = 5/13, so A is true. R is also true and is exactly why A holds. Both A and R are true, and R correctly explains A.
Q7 (Exam strategy tip). A common mistake in HOTS trigonometry questions is applying an identity meant for acute angles to a general angle without checking the domain, or forgetting to rationalise a surd in the final answer. Before submitting a proof, re-check that every identity used (like sin²θ+cos²θ=1 or 1+tan²θ=sec²θ) was valid for the specific range given in the question, and that the final simplified form matches what was asked — a numeric value, a proof, or a simplified expression.
Part of the Class 10 Maths Chapter 8 cluster: Solutions | Extra Questions (HOTS) | Revision Notes | Formulas Handbook | Class 10 Maths book page.
