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, $$x^2-y^2=(x-y)(x+y)$$ and this works for any x and y. So in this case $x=\cos^2\theta$ and $y=\sin^2\theta$

 

[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).