SOLUTION: Prove that sin^4 x - sin^2 x is equal to cos^4 x - cos^2 x

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Question 1190028: Prove that sin^4 x - sin^2 x is equal to cos^4 x - cos^2 x
Found 2 solutions by math_tutor2020, ikleyn:
Answer by math_tutor2020(3817) About Me  (Show Source):
You can put this solution on YOUR website!

I'll be using the pythagorean trig identity sin%5E2%28x%29%2Bcos%5E2%28x%29+=+1
That rearranges into sin%5E2%28x%29+=+1-cos%5E2%28x%29

From there, we can have this set of steps

sin%5E4%28x%29+-+sin%5E2%28x%29

%28sin%5E2%28x%29%29%5E2+-+sin%5E2%28x%29

%281-cos%5E2%28x%29%29%5E2+-+%281-cos%5E2%28x%29%29 Each sin^2 term is replaced with 1-cos^2

1-2cos%5E2%28x%29%2B%28cos%5E2%28x%29%29%5E2+-+%281-cos%5E2%28x%29%29 FOIL rule

1-2cos%5E2%28x%29%2Bcos%5E4%28x%29+-+1%2Bcos%5E2%28x%29

cos%5E4%28x%29-cos%5E2%28x%29

Therefore, sin%5E4%28x%29+-+sin%5E2%28x%29+=+cos%5E4%28x%29-cos%5E2%28x%29 is an identity. It's a true equation for all real numbers.

Answer by ikleyn(52803) About Me  (Show Source):
You can put this solution on YOUR website!
.
Prove that sin^4 x - sin^2 x is equal to cos^4 x - cos^2 x
~~~~~~~~~~~~~~ 

Based on Trigonometry identity sin^2(x) + cos^2(x) = 1, we have

    sin^2(x) = 1 - cos^2(x)  and  cos^2(x) = 1 - sin^2(x).


Using it, we have this one line chain of equivalent transformations


sin%5E4%28x%29+-+sin%5E2%28x%29 = sin%5E2%28x%29%2A%28sin%5E2%28x%29+-+1%29 = -sin%5E2%28x%29%2Acos%5E2%28x%29 = -%281-cos%5E2%28x%29%29%2A%28cos%5E2%28x%29%29 = cos%5E4%28x%29-cos%5E2%28x%29.


and the proof is completed at this point.

Solved.