I will now explain how to simplify trigonometric expressions, step by step, with resolute exercises, which will also help you to demonstrate whether certain expressions are equal.
In order to do this, it is necessary for you to know and master the fundamental equation of trigonometry, as well as other formulas where trigonometric ratios are related, in addition to the notable products. We will see it throughout this lesson.
Simplifying Trigonometric Expressions Exercises
As a general rule, the first step in simplifying a trig expression is to have only sinuses and cosines appear. You do this by using the formulas that relate the different trigonometric ratios (that’s why you need to control them). Each case will be different.
The next step is to operate with the remaining terms and apply formulas that allow us to continue grouping or deleting terms.
You’ll see it all much clearer if I explain it to you with resolute exercises, so let’s start by simplifying the following expression:
In this case, all the reasons that appear are sinuses and cosines, so in that sense we can do nothing else.
Therefore, we continue with the next step and operate, solving the squares of the parentheses.
To solve them, we use the remarkable products, more specifically the square of a sum and the square of a difference, of which I remind you of their formulas:
We apply and develop every remarkable product formula and it fits:
Now we see if we can simplify something and effectively, if you realize it, the term 2.sine.cosine is cancelled, as it appears adding and subtracting in the expression, so we remove them:
At this point, the fundamental equation of trigonometry comes into play for further simplification:
You may be wondering: How do I know when to use the fundamental equation of trigonometry to simplify trigonometric expressions?
I know because I have squared breasts and squared cosines. When I have squared sine and cosine, from the fundamental equation of trigonometry I can clear when it is worth sine squared, cosine squared or know that sine squared plus cosine squared equals 1.
In our expression, we have the sum of the sine squared plus the cosine squared repeated twice, resulting in 1:
For what we have left:
And the initial trigonometric expression is completely simplified.
As you can see, by applying the fundamental equation of trigonometry, we have simplified the expression in one step. That is why this equation is so important when it comes to simplification.
In the following examples, we will continue to use this equation, in addition to other formulas.
Here’s another example. Simplify the following trigonometric expression:
Don’t forget that the first step in simplifying a trig expression is to make only sinuses or cosines appear.
In this case, the tangent appears, so we have to replace the tangent by an expression in which sinuses and cosines appear, which we achieve with the following formula that relates these reasons:
We replace the tangent by this formula and it remains:
We already have only breasts and cosines, so the next step is to simplify operating. In this case we can operate on the fraction and that’s it:
And finally, it is clear that we only have to add the two cosines to simplify the expression:
When following the steps I am telling you, in each expression we will have to follow a different path, which sometimes is more clearly seen than others and depending on the reasons we have, we will have to use the formulas that relate the most appropriate trigonometric reasons.
Here’s another example. Simplifies the following trigonometric expression:
This case is a little different from the rest, in the sense that I always tell you that the first step is to make only breasts or cosines appear, but here first we are going to take another previous step.
As long as you have a sine, cosine or tangent squared minus 1, that is, these expressions:
You must develop it as the remarkable sum per difference product, which is equal to a difference of squares:
In this expression, in the numerator you have:
Which is the same as if it were:
Because 1 squared is 1. Therefore, when developing the remarkable product we are left with:
We realize this development within our expression:
And now, if you notice, the denominator is overridden by one of the factors of the numerator:
And this is why we have taken this previous step, developed the remarkable product of the square tangent minus 1, as a sum per difference. I was looking for the denominator to be overridden.
Now, we do apply the tangent formula so that only breasts and cosines appear in the expression:
We replace the tangent and we have it:
Now we’re going to operate. We multiply the cosine by each of the terms in the parenthesis:
In the first term, the cosines are cancelled:
And we have:
The trigonometric expression remains simplified.
As you can see, every case is different. I have given you a few steps to follow, but depending on each trig expression and your experience, it will tell you which formulas are most appropriate to use.
Exercises to demonstrate if the equations in trigonometric expressions are true
In addition to simplifying trigonometric expressions, there are also exercises in which you are asked to demonstrate whether the equality in trigonometric expression is true.
To carry out exercises of this type, we must simplify the expression, so the procedure is very similar to the previous section.
Let’s look at it with a couple of examples.
Demonstrate whether the following equality in trigonometric expression is true:
We start the same way as before, making the expression appear as breasts or cosines.
Therefore, by means of trigonometric formulas, we are going to replace and simplify in order to achieve this. First, we have a tangent, whose formula is:
Which we are now replacing in our expression:
Now let’s simplify the denominator of the first equation. This simplification is widely used to simplify trigonometric equations, so it is important that you understand it well.
From the fundamental equation of trigonometry:
We can clear the squared breast well and that’s it:
Or the cosine squared and that’s it:
In the denominator of the first equation you have:
So we can replace it with the squared sine and go from having two terms to only one, with which we can then operate:
This simplification is therefore widely used whenever you see it:
You know you can replace them with the square sine or the square cosine.
We continue to simplify. Now let’s go with the blotter squared. We know that:
We replace the formula of the squared secant in our expression:
And we’ve already reduced all the terms to breasts and cosines. Now we’re going to operate.
First we solve the square:
And now we’re operating on every fraction of it, remaining:
And we see that in fact, both the first and second members are equal to 1, so the equality in this case is true.
Let’s see another example. Demonstrate whether the following equality is true:
Let’s show that what it says on the first member is the same as what it says on the second member:
In the second limb there are only sinuses, therefore, the cosine of the first limb must be transformed into a sinus.
As I told you before, from the fundamental equation of trigonometry, we can clear the cosine square:
The square cosine as such does not appear in our expression. But because of the properties of the powers, when one power is raised to another, the exponents multiply.
Since we want cosine to appear squared, we can put it on and then raise it to the cube (which if you multiply the exponents we get back to the original 6):
Now we replace the square cosine with:
And we’re left with it:
Now, also according to the properties of the powers, when we have a multiplication of two powers with the same base, the exponents are added together, so we can put the parenthesis as a multiplication of powers, where one is squared and the other one is 1:
Now, the square bracket is a remarkable product:
So we developed it as such:
And now we multiply the two parentheses that we have left term by term:
By operating and grouping similar terms we are left with:
Which is the equality we wanted to demonstrate:
Therefore equality is true.