## Identify key patterns and apply mathematical reasoning to solve in a few tens of seconds

The chosen problem in this session is awkward and if you attempt to solve it through conventional deductions, it would be tedious and time consuming. You may even lose your way in confusion. Instead, if you are able to identify useful patterns and apply mathematical reasoning, solution would take just a few tens of seconds.

Before going ahead you may have look at 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 $\sin(B-C)\cos(A-D)$

$\hspace{22mm}+\sin(A-B)\cos(C-D)$

$\hspace{22mm}+\sin(C-A)\cos(B-D)$ is,

- $\displaystyle\frac{3}{2}$
- $0$
- $-3$
- $1$

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

We would expand each of the six compound angle $\sin$ functions and then simplify,

$\sin(B-C)\cos(A-D)$

$\hspace{22mm}+\sin(A-B)\cos(C-D)$

$\hspace{22mm}+\sin(C-A)\cos(B-D)$

$=(\sin B\cos C-\cos B\sin C)(\cos A\cos D+\sin A\sin D)$

$\hspace{5mm}(\sin A\cos B-\cos A\sin B)(\cos C\cos D+\sin C\sin D)$

$\hspace{5mm}(\sin C\cos A-\cos C\sin A)(\cos B\cos D+\sin B\sin D)$

$=\cos A\sin B\cos C\cos D+\sin A\sin B\cos C\sin D$

$\hspace{5mm}-\cos A\cos B\sin C\cos D-\sin A\cos B\sin C\sin D$

$\hspace{5mm}+\sin A\cos B\cos C\cos D+\sin A\cos B\sin C\sin D$

$\hspace{5mm}-\cos A\sin B\cos C\cos D-\cos A\sin B\sin C\sin D$

$\hspace{5mm}+\cos A\cos B\sin C\cos D+\cos A\sin B\sin C\sin D$

$\hspace{5mm}-\sin A\cos B\cos C\cos D-\sin A\sin B\cos C\sin D$.

These are the 12 four factor terms, six positive and six negative. Carefully going through the terms we identify cancellation pairs as,

Term 1 and term 7; Term 2 and term 12; Term 3 and term 9; Term 4 and term 6; Term 5 and term 11; Term 8 and term 10. All terms accounted for and cancelling each other, the final result is 0.

**Answer:** Option c: $0$.

*It is a purely deductive solution usually followed in schools. The difficulty in this problem arises at three levels—first, the two term expansions of six compound angle functions correctly needs concentration. Second—expansions of six two term factors resulting in 12 four factor terms again needs special attention; and the last—matching the 12 terms into cancelling pairs may be difficult unless the four angles in each of the 12 terms are ordered in the same manner. We have ordered it as A, B, C, D for all of the 12 terms.*

*Overall, this solution is impracticable in the context of an MCQ test. *

We will now present the elegant solution wholly based on key pattern identification and mathematical reasoning. For MCQ test, this is the only practicable solution for this problem.

#### Second solution: Problem analysis, Key pattern identification and Solution by mathematical reasoning

Each of the three product terms will expand to 4 unique terms because of the unique angle and function distribution. No term will be cancelled out within each product result and after expansion then, total number of terms will be 12.

The important pattern in these 12 terms identified is—each of the 12 terms consists of four unique factors of $\sin$ and $\cos$ *using all the four angles*.

Continuing our **deductive reasoning** it is obvious then,

All these 12 compound terms, each with four factors in $\sin$ and $\cos$, must cancel out against each other to generate the

final result free of any variable.

Finally we reason that—if six of these twelve terms cancel out against the other six, the final result can only be 0, not any non-zero integer.

**Answer:** Option b: $0$.

**Key concepts and techniques used: ** *Problem analysis -- Key pattern identification -- Deductive reasoning -- Mathematical reasoning -- Solving in mind.*

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.

### Resources on Trigonometry and related topics

You may refer to our useful resources on Trigonometry and other related topics especially algebra.

#### Tutorials on Trigonometry

**Basic and rich concepts in Trigonometry and its applications**

**Basic and Rich Concepts in Trigonometry part 2, proof of compound angle functions**

**Trigonometry concepts part 3, maxima (or minima) of Trigonometric expressions**

#### General guidelines for success in SSC CGL

**7 steps for sure success in SSC CGL tier 1 and tier 2 competitive tests**

#### Efficient problem solving in Trigonometry

**How to solve a difficult SSC CGL level problem in a few reasoned steps, Trigonometry 10**

**How to solve not so difficult SSC CGL level problem in a few light steps, Trigonometry 9**

**How to solve a difficult SSC CGL level problem in a few conceptual steps, Trigonometry 8 **

**How to solve not so difficult SSC CGL level problem in a few light steps, Trigonometry 7**

**How to solve a difficult SSC CGL level problem in few quick steps, Trigonometry 6**

**How to solve a School Math problem in a few direct steps, Trigonometry 5**

**How to solve difficult SSC CGL level School math problems in a few quick steps, Trigonometry 5**

**How to solve School Math problem in a few steps and in Many Ways, Trigonometry 4**

**How to solve a School Math problem in a few simple steps, Trigonometry 3**

**How to solve difficult SSC CGL level School math problems in a few simple steps, Trigonometry 4**

**How to solve difficult SSC CGL level School math problems in a few simple steps, Trigonometry 3**

**How to solve School math problems in a few simple steps, Trigonometry 2**

**How to solve School math problems in a few simple steps, Trigonometry 1**

**A note on usability:** The *Efficient math problem solving* sessions on **School maths** are **equally usable for SSC CGL aspirants**, as firstly, the "Prove the identity" problems can easily be converted to a MCQ type question, and secondly, the same set of problem solving reasoning and techniques have been used for any efficient Trigonometry problem solving.

#### SSC CGL Tier II level question and solution sets on Trigonometry

**SSC CGL Tier II level Solution Set 18 Trigonometry 4, questions with solutions**

**SSC CGL Tier II level Question Set 18 Trigonometry 4, questions with answers**

**SSC CGL Tier II level Solution set 12 Trigonometry 3, questions with solutions**

**SSC CGL Tier II level Question set 12 Trigonometry 3, questions with answers**

**SSC CGL Tier II level Solution set 11 Trigonometry 2**

**SSC CGL Tier II level Question set 11 Trigonometry 2**

**SSC CGL Tier II level Solution set 7 Trigonometry 1**

**SSC CGL Tier II level Question set 7 Trigonometry 1**

#### SSC CGL level question and solution sets in Trigonometry

**SSC CGL level Solution set 82 on Trigonometry 8**

**SSC CGL level Question set 82 on Trigonometry 8**

**SSC CGL level solution set 77 on Trigonometry 7**

**SSC CGL level question set 77 on Trigonometry 7**

**SSC CGL level Solution Set 65 on Trigonometry 6**

**SSC CGL level Question Set 65 on Trigonometry 6**

**SSC CGL level Solution Set 56 on Trigonometry 5**

**SSC CGL level Question Set 56 on Trigonometry 5**

**SSC CGL level Solution Set 40 on Trigonometry 4**

**SSC CGL level Question Set 40 on Trigonometry 4**

**SSC CGL level Solution Set 19 on Trigonometry**

**SSC CGL level Question set 19 on Trigonometry**

**SSC CGL level Solution Set 16 on Trigonometry**

**SSC CGL level Question Set 16 on Trigonometry**

**SSC CGL level Question Set 2 on Trigonometry**

**SSC CGL level Solution Set 2 on Trigonometry**

#### Algebraic concepts

**Basic and rich Algebraic concepts for elegant solutions of SSC CGL problems**

**More rich algebraic concepts and techniques for elega****n****t solutions of SSC CGL problems**