[AM_20150621TGCARF] Sum of sin and cos the cost of tan?

Question

Introduction

     This is a bonus Maths II (rough equivalent of Additional Mathematics syllabus) question from a test by Hwa Chong Institution (HCI) this year 2015.  Independent schools in Singapore like HCI are free to set their own curricula, but they usually end up covering slightly more than the mainstream curriculum, since their students also take the national examinations.  For their internal tests and exams, they can set bonus questions.  These are harder but optional questions that students can attempt and if they are successful, the bonus marks can be added to their normal marks.  It is also OK not to attempt the bonus questions.  That gives students the choice and opportunity to stretch their minds, but they are not penalised if they are unable to solve the bonus questions.     In this article, I present two approaches to tackling this question.  Let us review some important formulas first.

Some Useful Formulas

Solution 1  (via the Pythagorean Identity)

  

Solution 2  (via R-formula)

Reflections / Extension

     Here is another HCI question on trigonometry that involves sine, cosine and tangent.

Heuristics Used

H04. Look for pattern(s)

H05. Work backwards

H09. Restate the problem in another way

H10. Simplify the problem

H11. Solve part of the problem

H13* Use Equation / write a Mathematical Sentence

Suitable Levels

* GCE ‘O’ Level Additional Mathematics

* other syllabuses that involve trigonometry