How Do You Simplify Trigonometric Expressions Using Identities?

[Olly_price]

I'm teaching maths to myself so I don't really have anywhere else to go for an explanation other than here, so I apologise if this seems simple.

How do you get from:

(cos^2θ + sin^2θ)(cos^2θ - sin^2θ)

to

cos^4θ - sin^4θ

NOTE: cos^2θ is shorthand for (cosθ)^2 as is with all the other ones as well.

The question could also be asked in reverse (how do I factorise cos^4θ - sin^4θ)

Please bear in mind that I am teaching maths to myself, so it's pretty useless if you don't explain every mathematical step.

 

[coolul007]

It is just algebra if cos = a, and sin = b, then cos^2 = a^2, and sin^2 = b^2

 

[Mentallic]

And to take what coolul007 said a little further, it's a difference of two squares.

In general, [tex]x^2-y^2=(x-y)(x+y)[/tex] and this works for any x and y. So in this case [itex]x=\cos^2\theta[/itex] and [itex]y=\sin^2\theta[/itex]

 

[chiro]

Hey Olly_price and welcome to the forums.

You should learn about the distributive law and expand out the (x-y)(x+y) in terms of x's and y's and you will end up showing the formula Mentallic described above.

This will help you if you come across more complicated expressions where you need to show a similar kind of (example (x-y)(x-y)(x-y) expanded).

 

[CellCoree]

I understand that learning new concepts and equations can be challenging. I will do my best to explain the process of factoring radian equations in a clear and concise manner.

To begin, let's review the basic trigonometric identities that will be used in this process:

1. Pythagorean Identity: cos^2θ + sin^2θ = 1

2. Difference of Squares Identity: a^2 - b^2 = (a + b)(a - b)

Now, let's look at the equation (cos^2θ + sin^2θ)(cos^2θ - sin^2θ). We can apply the Pythagorean Identity to the first set of parentheses, which will give us:

1(cos^2θ - sin^2θ)

Next, we can apply the Difference of Squares Identity to the second set of parentheses, which will give us:

1(cos^2θ - sin^2θ)(cos^2θ + sin^2θ)

We can now see that the two sets of parentheses are identical, so we can simplify the equation to just:

1(cos^2θ - sin^2θ)^2

To factor this further, we can use the Difference of Squares Identity again, which will give us:

1(cosθ - sinθ)(cosθ + sinθ)(cosθ - sinθ)(cosθ + sinθ)

This can be simplified to:

1(cos^2θ - sin^2θ)(cos^2θ - sin^2θ)

Finally, we can apply the Pythagorean Identity once more to get:

1(cos^4θ - sin^4θ)

In summary, the process of factoring radian equations involves using basic trigonometric identities to simplify the equation and then applying the Difference of Squares Identity to factor it further. I hope this explanation helps in your understanding of this concept. Keep up the good work in teaching yourself mathematics!