Trigonometric Identity Proving is a common question type that is included in the O-Level Additional Math syllabus. The mention of “trigo proving” would often cause even the top secondary school students to break out in cold sweat. This is because, unlike most A-Math (O-level) topics, trigonometry proving questions do not have a standard “plug and play” method of solving. Every question is a new puzzle for which the students have to find a route from start to end. Very often, students adopt a 走一步看一步 (Directly translated as: Walk one step, watch one step) approach to solving these questions.
Although every question is unique, there are actually several “rules of thumb” for which students can follow such that they do not get lost. Here, I shall distill some precious tips to help students conquer Trigo proving.
Tip 1) Always Start from the More Complex Side
To prove a trigonometric identity, we always start from either the left hand side (LHS) or the right hand side (RHS) and apply the identities step by step until we reach the other side. However, smart students always start from the more complex side. This is because it is a lot easier to eliminate terms to make a complex function simple than to find ways to introduce terms to make a simple function complex.
Example Q1) Prove the identity tan⁴x = sec²x (tan²x-1)+1
Approach: It would be wise to start proving this from the right hand side (RHS) since it is more complex.
Tip 2) Express everything into Sine and Cosine
To both sides of the equation, express all tan , cosec , sec and cot in terms of sin and cos . This is to standardize both sides of the trigonometric identity such that it is easier to compare one side to another.
Tip 3) Combine Terms into a Single Fraction
When there are 2 terms on one side and 1 term on the other side, combine the side with 2 terms into 1 fraction after making their denominators the same.
Tip 4) Use Pythagorean Identities to transform between sin²x and cos²x
Pay special attention to addition of squared trigonometry terms. Apply the Pythagorean identities when necessary. Especially sin²x+cos²x=1 since all the other trigo terms have been converted into sine and cosine. This identity can be used to convert into and vice versa. It can also be used to remove both by turning it into 1.
Tip 5) Know when to Apply Double Angle Formula (DAF)
Observe every trigonometric term in the question. Are there terms with angles that are 2 times of another? If there are, be ready to use DAF to transform them into the same angle. For example, if you see sinθ and cot(θ/2) in the same question, you have to use DAF since θ is 2 times of (θ/2).
Tip 6) Know when to Apply Addition Formula (AF)
Observe the angles in the trigonometric functions. Are there summations of 2 different terms in the same Trigonometric term? If the answer is yes, apply the addition formula (AF).
Tip 7) Good Old Expand/ Factorize/ Simplify/ Cancelling
Many students hold on to the false belief that every single trigonometry proving question require the use of trigonometric identities from the formula sheet. Whenever they get stuck, they resort to staring blindly at the formula sheet and praying that the answer will magically “jump out” at them. More often than not, the miracle does not happen. This is because most proving questions revolve majorly around good old expansion, factorization, simplification and cancelling of like terms. In fact, some proving questions do not even require student to use any trigonometry rules at all.
In addition, always look out for opportunities to apply quadratic identities that you have mastered in Secondary 2:
Tip 8) Take one Step, Watch one step.
Proving trigonometry functions is an art. There are often several ways to get to the answer. Naturally, some methods are more elegant and short while other methods are crude, massive and ugly. However, the key point to note is that whichever way we take, as long as we can get to the final destination, we will get the marks.
Some students would spend a long time staring at the question and attempt to work out the entire solution in their Pentium 9999 brain processor. I applaud them for their heroic attempt. Unfortunately, most of them run out of RAM and switch off before the question is completed. On the other hand, there are “Kan cheong spiders” who would immediately pick up their pens and start scribbling down random steps without thinking. These students would end up wasting time heading towards nowhere and have to restart a few times.
The most experienced students would balance between both. They would spend some time to get their bearings and courageously take their first step. After every one or two steps, they would re-analyze their proximity to the final destination before deciding on the next step.
Tip 9) When Desperate… Pretend!
Disclaimer: Only use this tactic if you find yourself stuck half way during the trigo proving process in an examination (with the clock ticking away) and you do not want to jeopardize the rest of the paper. Since you are stuck mid way, simply complete the question by pretending that you have proved the identity. From your current step, jump straight to the final step and then write (=RHS (Proven)). After the exam, do remember to visit the nearest Temple/Church/Mosque to pray hard that the marker is either blind or compassionate enough to give you the benefit of the doubt and award you the marks.
*Note: This working is actually wrong because there are several missing steps between the second step and last step. However, this working illustrates the point that when you are stuck and desperate during the O-level examination, you should still “pretend” to prove the question by writing the last step down.
For the actual full solution of this question, scroll down to the bottom.
Tip 10) Practice! Practice! Practice!
Proving trigonometric function becomes a piece of cake after you have conquered a massive number questions and expose yourself to all the different varieties of questions. There are no hard and fast rule to handling O-level trigonometry proving questions since every question is like a puzzle. But once you have solved a puzzle before, it becomes easier to solve the same puzzle again.
Tip 11) Do not try to Prove a Question that says “Solve”!
After practicing a massive number of proving questions, some students develop a robotic tendency to prove LHS = RHS whenever they see an equation with trigonometric functions. Even when they encounter a question that says "Solve the trigonometry equation..."... Do read the question carefully! If the question expects you to “Solve”, do not try to prove it! You can try proving till the cows come home but you will never be able to do it.
Example Q11) Solve the equation 5 cosecx - 3 sinx = 5 cotx
Approach: This is a “solve question” (i.e. find the values of x ). DO NOT attempt to prove it because you cant!
* Solutions to the examples in the above Trigo Proving questions can be downloaded here
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