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← 2020 HSC Extension 1 paper Question 13(c)

Variation 3 · answer guide

Inverse trigonometric functions — Sec of arctan

Question

Suppose and for .

(i) Show that .

(ii) Using part (i), or otherwise, show that .

Graph of $g(x)=\sqrt{1+x^2}$ for all real $x$.
Graph of for all real .

This practice question was inspired by Question 13(c) in the 2020 NSW HSC Mathematics Extension 1 examination, © NESA. It is an original TestMum variation that practises the same mathematical idea, not an official NESA question.

Step by step

  1. Set up the solution: use the restricted domain and range that make the inverse function single-valued.(i) Chain rule: .
  2. Simplify the previous line carefully, keeping exact values where possible.With and (principal value ), .
  3. Simplify the previous line carefully, keeping exact values where possible.Differentiate : .
  4. Simplify the previous line carefully, keeping exact values where possible.Hence .
  5. Finish the calculation, then check that the result meets the question’s conditions.(ii) So on , and is constant. Test : and .
  6. State the final answer clearly in the form the question requested.Therefore for all real .

Common mistake: changing so many features that the new example no longer tests the same mathematical idea.

Verification: Right-triangle: opposite , adjacent , hypotenuse , so .

Any mark labels are a TestMum study aid, not an official NESA marking allocation.

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