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How to solve a School Math problem in a few simple steps, Trigonometry 2

Intelligent School Math solutions in a few steps, Trigonometry 2

Intelligent solution to the second school math problem

The student's intelligence is used for solving the second school math problem in far simpler steps compared to the complex conventional method of solution.

Many times we find at high school level, math problems are solved following a long series of steps. This is what we call conventional approach to solving problems.

This approach not only involves usually a large number of steps, in most cases, the steps themselves introduce a higher level of complexity.

These two aspects of conventional problem solving together result in what we call costly and inefficient problem solving.

In school math or real life problems, this is the path commonly followed by most. In school math, among other topics we find this form of solution abundantly in case of problems of the type,

Prove that, "Some expression " = "Some other expression".

In school math terminology, the "Some expression" is called LHS (short form of Left Hand Side) and the "Some other expression" as RHS (short form of Right Hand Side). This type of problems occurs aplenty in Elementary Trigonometry of proving Identities.

We have covered one such example already, This is the second example.

These long and mostly complex solutions use a conventional approach of going towards the solution from LHS to RHS (or initial state to goal state) through many steps using the expansion of the LHS expression and then simplification or consolidation of the numerous expanded terms towards the form of expression on the right hand side, that is, the RHS.

This approach has two important disadvantages,

  1. Not only does this approach take considerable amount of time and effort, but because of large number of steps and increased complexity, chances of error is much higher in this approach.
  2. This mechanical approach relies heavily on manipulation of terms using low level mathematical constructs without using the problem solving abilities of the student. In fact, if students follow only this approach for solving problems, they may tend to become used to mechanical and procedural thinking suppressing their inherent creative and innovative out-of-the-box thinking abilities.

Let us go through the process of solving another Trigonometric Identity problem to appreciate the difference between the conventional approach and the problem solver's intelligent approach. The difference is always significant and measurable.

Problem example

Prove the identity:

$\displaystyle\frac{sin\theta}{1 - cos\theta} + \displaystyle\frac{tan\theta}{1 + cos\theta} = sec\theta{cosec\theta} + cot\theta$

First try to solve this problem yourself and then only go ahead. You might be able to reach the elegant solution to this problem yourself.

Conventional solution

Usually a conventional approach in this case will involve a long and complex deduction process involving a large amount of mathematical manipulation.

We combine the two terms in a brute-force method (there can be other such conventional methods) to transform the LHS as,

\begin{align} & \displaystyle\frac{sin\theta(1 + cos\theta) + tan\theta(1 - cos\theta)}{(1 - cos\theta)(1 + cos\theta)} \\ & = \displaystyle\frac{sin\theta + sin\theta{cos\theta} + tan\theta - sin\theta}{1 - cos^2\theta} \\ & = \displaystyle\frac{sin\theta{cos\theta} + tan\theta}{sin^2\theta}  \\ & = cosec^2\theta(sin\theta{cos\theta} + tan\theta) \\& = cosec\theta{cos\theta} + cosec\theta{sec\theta} \\ & = sec\theta{cosec\theta} + cot\theta\end{align}

In this method we went through a straightforward approach of combining the two fraction terms in the LHS, and going though carefully the simplification process relying on our knowledge of conversion of trigonometric terms so that we can move towards the end state of the given expression with certainty and no error.

As such there is nothing wrong with this conventional straightforward approach.

On second thoughts though, we find we have to go through a lot of trigonometric procedures and the complexity level of the deductions is high.

In the discipline of Efficient Math Problem Solving we emphasize repeatedly, that

Always use your intelligent problem solving ideas more and the routine mathematical procedures less in solving any math problem.

This is broadly our Less facts and more procedures approach that we would deal with in detail later. Nevertheless, each of the efficient math problem solving examples actually use this more abstract problem solving approach. Formally we state this Less facts and more procedures approach as,

For solving any problem, use least amount of unconnected facts and more of interlinked concepts and efficient problem solving procedures.

In maths problem solving, this approach boils down to the strong recommendation highlighted by many eminent mathematicians among others,

Do more problem solving and less maths.

Efficient solution in a few steps

Instead of this conventional approach, the very first step that you must take as usual is to analyze the problem.

In any problem solving, math or otherwise, this must be the first step. You must start with analyzing the problem statement.

Without the first step of Problem analysis, no efficient problem solving is possible.

A corollary,

In competitive exams, and also in competitive work environment, the first step of problem analysis is crucial for success.

The better and quicker you are able to analyze a problem, the faster you would reach the desired solution.

Problem analysis

As a first step of analysis in such math problems, we follow the End State analysis approach invariably and compare the desired end state expression with the given initial state expression for detecting similarities or dissimilarities.

In this problem we observe, though the terms of the given expression are fractions involving denominators, the end state goal expression on the RHS does not have any fractions. This is the significant dissimilarity that would guide us to our main strategy to deal with the problem solution elegantly.

Key information discovery

Thus we form our goal as transforming each of the two terms in the LHS to eliminate the denominators as quickly and as elegantly as possible.

This objective is further strengthened when we observe both the denominators in the form of $x + y$ or $x - y$.

This is a valuable information to us because of our knowledge of the frequently used Technique of Rationalization using the important concept,

$(x + y)(x - y) = x^2 - y^2$.

While rationalizing a fraction term (usually this technique is applied in rationalizing surd expressions involving terms in the denominator such as $(\sqrt{2} + 1$), we multiply both the numerator and the denominator of the fraction with a suitable expression such that the denominator is highly simplified. Most often this simplification happens using the concept of $x^2 - y^2 = (x + y)(x - y)$.

In Trigonometry, rationalization of this form further uses the most fundamental concept,

$sin^2\theta + cos^2\theta = 1$,

Or, $1 - cos^2\theta = sin^2\theta$,

Or, $1 - sin^2\theta = cos^2\theta$.

Armed with this result of our analysis along with the concept of rationalization technique, we proceed to rationalize each term of the LHS to get,

$LHS = \displaystyle\frac{(1 + cos\theta)sin\theta}{1 - cos^2\theta} $

$\hspace{25mm} + \displaystyle\frac{(1 - cos\theta)tan\theta}{1 - cos^2\theta}$

Or, $LHS = cosec\theta(1 + cos\theta) $

$ \hspace{25mm} + sec\theta{cosec\theta}(1 - cos\theta)$.

At this stage, we have used the concept, $1 - cos^2\theta = sin^2\theta = \displaystyle\frac{1}{cosec^2\theta}$.

Or, $LHS = cosec\theta + cot\theta $

$ \hspace{25mm} + sec\theta{cosec\theta} - cosec\theta$

$ \hspace{20mm} = RHS$.

This is by far the quickest way to reach the solution in just three steps and is based fully on examining and using the dissimilarity between the final state and the initial state and deciding that rationalization would make the states (or expressions) more similar.  

Note: In this solution, we have skipped a few trivial steps that you may not be allowed to skip in your formal prescribed school procedure. Saving in that case won't be so much as in reduced number of steps, rather the saving would be in following a less complex path to the solution resulting in saving in time and reducing chances of error.

Deductive reasoning

On comparison of the RHS and LHS we find the given expression having terms with denominators while the goal state expression is without any fraction term involving denominators.

Furthermore, inspecting the denominators of the fraction tems, we observe a pattern in each that should enable us to easily eliminate the denominators by rationalization.

This is where we use our Pattern recognition technique.

Thus we form our first conjecture as,

The immediate task is to rationalize each term of the given expression using concepts $x^2 - y^2 = (x + y)(x - y)$ together with $1 - cos^2\theta = sin^2\theta$.

At the second stage, rationalizing and transformation of the given terms using simple trigonometric relations quickly take us to the desired goal expression.

End note: This is not the only example of this type. You would find very large number of this and other type of problems where you would invariably be able to improve upon the time and effort in conventional solution.

Our recommendation: Always think: is there any other shorter better way to the solution?

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