Simple Trig Proof: prove that sin^4θ-cos^4θ=1-2cos^2θ.

TheAgileWarrior249

New member
Joined
Nov 4, 2018
Messages
3
Use the Pythagorean Trig Identity, sin^2θ+cos^2θ=1, to prove that sin^4θ-cos^4θ=1-2cos^2θ.

With the advice of my teacher, I started by making a chart. One side is "sin^4θ - cos^4θ", and the other is "1 - 2cos^2θ". He said I'm not supposed to plug in any numerical values, but I'm not sure what to do next.
 
Use the Pythagorean Trig Identity, sin^2θ+cos^2θ=1, to prove that sin^4θ-cos^4θ=1-2cos^2θ.

With the advice of my teacher, I started by making a chart. One side is "sin^4θ - cos^4θ", and the other is "1 - 2cos^2θ". He said I'm not supposed to plug in any numerical values, but I'm not sure what to do next.
sin^2(θ) + cos^2(θ)=1

sin^2(θ) = 1 - cos^2(θ)

[sin^2(θ)]^2 = [1 - cos^2(θ)]^2

sin^4(θ) = 1 - 2*cos^2(θ) + cos^4(θ)

continue...
 
Use the Pythagorean Trig Identity, sin^2θ+cos^2θ=1, to prove that sin^4θ-cos^4θ=1-2cos^2θ.

With the advice of my teacher, I started by making a chart. One side is "sin^4θ - cos^4θ", and the other is "1 - 2cos^2θ". He said I'm not supposed to plug in any numerical values, but I'm not sure what to do next.

It might help to recall that \(\displaystyle a^2-b^2 = (a+b)(a-b)\)

or in particular that
\(\displaystyle a^4-b^4 = (a^2+b^2)(a^2-b^2)\)
 
Use the Pythagorean Trig Identity, sin^2θ+cos^2θ=1, to prove that sin^4θ-cos^4θ=1-2cos^2θ.

With the advice of my teacher, I started by making a chart. One side is "sin^4θ - cos^4θ", and the other is "1 - 2cos^2θ". He said I'm not supposed to plug in any numerical values, but I'm not sure what to do next.

It isn't clear exactly what your teacher is asking you to do, and what you did. There are many ways to prove any identity.

My impression is that you were taught to put each side of the identity you are to prove at the top of a column, and then apply known identities to transform one to the other, or to transform each to the same thing (in a reversible way). If so, then I might start on the left side by factoring as a difference of squares (as has been mentioned); one factor then is 1. Then you might either continue with that, replacing sin^2(θ) with 1 - cos^2(θ), or do something similar on the right. Most teachers would prefer you to work on one side, though that isn't essential if you know what you are doing.

Can you show your work, or an example from your teacher, so we can be more sure what style of proof you are looking for?
 
Top