How to Solve Complex Trigonometric Equations Using Substitution Method

The substitution Method can be used to solve a variety of problems and Trigonometric Equations is one of them. Read on to find out how.

SSC CGL and Railway Recruitment Board are some of the competitive exams where Trigonometric Equations is a part of the syllabus. However this is one topic that scares a lot of us because of the number of identities involved but we have you covered here too! The substitution method is an answer to your prayers.
So in the 17th smart method in this series of blog posts, we will discuss how to use the substitution method to solve complex trigonometric equations.

Trigonometric Equations

When in school the one topic that gave a lot of us nightmares was Trigonometry. The never ending, super confusing list of formulas to solve trigonometric equations was a mystery that we always struggled to solve since a lot of us could never really understand how one complex expression lead to another super complex expression! Your prayers are finally answered because the substitution method to solve trigonometric equations will relive you of this misery.

Example of a Question on Trigonometric Equations

The value of  (cos³θ + sin³θ)/(cosθ + sinθ) + (cos³θ – sin³θ)/(cosθ – sinθ) is equal to –
1) -1      2) 1      3) 2      4) 0      5) -2
Conventional Method to Solve Trigonometric Equations

Even though you may dread at the thought of it, you probably remember what we were taught in school – use algebraic identities with trigonometric ones.
Using the above algebraic identities we to the given trigonometric expression we get

Step 1:
Step 2:
Step 3:
Step 4:
Honestly in an exam where you are racing against time, this is way too complicated.

Smart Method to Solve Trigonometric Equations

Welcome to the world of smart methods where we will use the substitution method to solve this expression and reach the answer in about 15 seconds unlike the conventional method which takes 120 seconds.
We will substitute the value of 𝛳 with any standard numeric value instead of using algebraic and trigonometric identities and making it more complex. If you read the question carefully that the answer is independent of the value of 𝛳.
Let us say 𝛳 = 0

Step 1:
Step 2:
(1+0)/(1+0) + (1-0)/(1-0)

Step 3:
1+1 = 2

Do watch the video below to see how our expert faculty uses substitution method to solve such complicated trigonometric equations!

You must try to solve these trigonometric equations by using substitution method-

Question 1: The value of  [1/ (1 + tan²θ)] + [1/ (1 + cot²θ)] is
1) 1/4      2) 1      3) 2      4) 1/2      5) None of these
Question 2: The value [1/ (cosecθ – cotθ)] –  (1/ sinθ) is
1) 1      2) cotθ      3) cosecθ     4) tanθ     5) None of these
Remember to leave your answer in the comments section along with the time you took to solve the trigonometric equation.
Stay tuned to simplify many more such problems the smart way! Also start practicing the smart way!

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