**The medium level problem can be solved in just a few steps if you discover the concept based key pattern**

The chosen problem this time is not particularly difficult, and on most occasions it will be solved conventionally. But driven by the general objective of equalizing the denominators, if the lightly hidden key pattern is discovered, everything becomes easy and straightforward with solution only a few light steps away.

In this session we will showcase a not so difficult problem. With such problems on most occasions we take the conventional path. The first solution presented will thus be the deductive conventional one.

This is how we think and act, conventionally, most of the times.

In contrast to the conventional solution, we will present the elegant solution next. It will highlight, how driven by a general problem solving objective, a key pattern and conceptual method to use the pattern can be discovered for solving the problem in only a few easy steps.

The *concepts and techniques used in the elegant solution are easy and straightforward*, and anyone can think in this way, saving valuable seconds in comparison to the conventional way to the solution.

Before going ahead you should refer to our concept tutorials on Trigonometry,

**Basic and Rich Trigonometry concepts and applications**

**Basic and Rich Trigonometry concepts part 2, Compound angle functions**

*Trigonometry concepts part 3, maxima and minima of trigonometric expressions.*

**Chosen Problem.**

The value of $sec \theta\left(\displaystyle\frac{1+sin \theta}{cos \theta}+\displaystyle\frac{cos \theta}{1+sin \theta}\right) - 2tan^2 \theta$ is,

- 4
- 0
- 2
- 1

**First solution: Conventional approach with deductive steps**

If the two balanced terms within the brackets are combined, the numerator is formed as sum of two squares. The expression being not very complex and balanced in nature, it should not be difficult to simplifiy the sum of squares numerator. This is the conventional thinking adopted most of the times for this problem.

The given expression is,

$E=sec \theta\left(\displaystyle\frac{1+sin \theta}{cos \theta}+\displaystyle\frac{cos \theta}{1+sin \theta}\right) - 2tan^2 \theta$

$=sec \theta\left(\displaystyle\frac{(1+sin \theta)^2 +cos^2 \theta}{cos \theta(1+sin \theta)}\right) - 2tan^2 \theta$

$=sec \theta\left(\displaystyle\frac{1 + 2sin \theta +sin^2 \theta +cos^2 \theta}{cos \theta(1+sin \theta)}\right)-2tan^2 \theta$

$=sec \theta\left(\displaystyle\frac{2 + 2sin \theta}{cos \theta(1+sin \theta)}\right) - 2tan^2 \theta$

$=2sec \theta\left(\displaystyle\frac{(1 + sin \theta)}{cos \theta(1+sin \theta)}\right)-2tan^2 \theta$

$=2sec^2 \theta - 2tan^2 \theta$

$=2$.

**Answer:** Option c: 2.

*It is a purely deductive solution usually followed in schools.*

We will now present the elegant solution where hidden simplifying patterns are discovered and used, thus simplifying the expression significantly in just a few steps avoiding the cumbersome, time-consuming squaring and simplification process.

#### Second solution: Elegant Solution: Denominator equalization by denominator rationalization

Though simple in nature, in any problem involving simplification of multiple additive fractional terms, * denominator equalization technique* always achieves an elegant solution. In this case also, we look for an way to equalize the denominators of the two terms.

A second great result we have experienced so often,

If you look for some special result in particular, chances will be high that you will get it.

Conversely,

If you don't look for a new opportunity, you will never be able to discover it.

In this problem, when we searched for an way to equalize the denominators, the lightly hidden opportunity of rationalizing the denominator of the second term revealed itself.

Let us show how.

The given expression,

$E=sec \theta\left(\displaystyle\frac{1+sin \theta}{cos \theta}+\displaystyle\frac{cos \theta}{1+sin \theta}\right)-2tan^2 \theta$

$=sec \theta\left(\displaystyle\frac{1+sin \theta}{cos \theta}+\displaystyle\frac{cos \theta(1-sin \theta)}{1-sin^2 \theta}\right)-2tan^2 \theta$

$=sec \theta\left(\displaystyle\frac{1+sin \theta}{cos \theta}+\displaystyle\frac{cos \theta(1-sin \theta)}{cos^2 \theta}\right)-2tan^2 \theta$

$=sec \theta\left(\displaystyle\frac{1+sin \theta}{cos \theta}+\displaystyle\frac{1-sin \theta}{cos \theta}\right)-2tan^2 \theta$

$=2sec \theta\left(\displaystyle\frac{1}{cos \theta}\right)-2tan^2 \theta$

$=2sec^2 \theta - 2tan^2 \theta$

$=2$.

**Answer:** Option c: 2.

**Key concepts used: Deductive reasoning **to attach priority to denominator equalization as a means to elegant simplification and identification and application of subsequent problem solving steps in a chained sequence

**--**--

*Denominator equalization**--*

**Key patttern identification***--*

**Denominator rationalization***--*

**Basic trigonometry concepts***--*

**Basic algebra concepts***--*

**Rich algebra techniques**

**Efficient simplification -- Many ways technique.**The squaring and simplification of the denominators is completely avoided which is the hallmark of problem solver's elegant solution.

Compare the shortcomings of the two solutions and identify the reasons behind the shortcomings.

**Important to note**

Elegant and efficient solutions are based on problem analysis, deductive reasoning and use of problem solving strategies and techniques. If you attempt to solve a problem using concepts in a few steps, gradually you will develop the skillset in reaching an elegant solution for most problems easily.

It is in a way, a habit of thinking in new ways.

Conversely, if you always follow the conventional path to the solution of a problem, you will never know how to think new and how to solve problems elegantly and efficiently.

#### Special characteristics of efficient trigonometry problem solving

- To achieve elegant and efficient solutions in Trigonometry consistently,
**role of basic and rich algebraic concepts and techniques is critically important.** - While solving the problem in two different ways, we have applied the problem solving skill improvement
that tested our skill in solving a problem in multiple ways as well as gave us the opportunity to compare the multiple solutions with each other.**Many ways technique**

Even simple things need to be made simpler. Only then more difficult things can be made simple with ease.

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