how would you prove that
(cos^2x-sin^2x)(1-cos^2xsin^2x)=
(cos^6-sin^6)?
On the left
cos^2 x - cos^4 x sin^2 x - sin^2 x + cos^2 x sin^4 x
= cos^2 x - cos^4 x (1 - cos^2 x) - sin^2 x + sin^4 x (1 -sin^2 x)
= cos^2 x -cos^4 x + cos^6 x - sin^2 x + sin^4 x - sin^6 x
= cos^6 x - sin^6 x
+ cos^2 x - sin^2 x + sin^4 x - cos^4 x
= cos^6 x - sin^6 x
+ cos^2 x - sin^2 x
+ (sin^2 x - cos^2 x)(sin^2 x + cos^2 x)
= cos^6 x - sin^6 x
+ cos^2 x - sin^2 x
- (cos^2 x - sin^2 x)(sin^2 x + cos^2 x)
= cos^6 x - sin^6 x
- (cos^2 x - sin^2 x)(-1 + sin^2 x + cos^2 x)
but
-1 + sin^2 x + cos^2 x = 0
Wow, that was a good one !
To prove that (cos^2x - sin^2x)(1 - cos^2xsin^2x) equals (cos^6x - sin^6x), we can simplify both sides of the equation and then assess if they are equal to each other.
Let's start by simplifying the left-hand side (LHS) of the equation:
(cos^2x - sin^2x)(1 - cos^2xsin^2x)
Using the identity cos^2x - sin^2x = cos2x, we can rewrite the LHS as:
cos2x(1 - cos^2xsin^2x)
Now, let's simplify the right-hand side (RHS) of the equation:
cos^6x - sin^6x
Using the identity cos^2x = (1 + cos2x)/2 and sin^2x = (1 - cos2x)/2, we can rewrite the RHS as:
[(1 + cos2x)/2]^3 - [(1 - cos2x)/2]^3
Now, let's expand these expressions:
[(1 + cos2x)/2]^3 = (1 + 3cos2x + 3cos^2(2x) + cos^3(2x))/8
[(1 - cos2x)/2]^3 = (1 - 3cos2x + 3cos^2(2x) - cos^3(2x))/8
Substituting these expanded expressions back into the equation, we have:
(1 + 3cos2x + 3cos^2(2x) + cos^3(2x))/8 - (1 - 3cos2x + 3cos^2(2x) - cos^3(2x))/8
Simplifying further, the terms with opposite signs cancel each other out:
(3cos2x - 3cos2x + 3cos^2(2x) + 3cos^2(2x) + cos^3(2x) - (-cos^3(2x)))/8
Simplifying once more:
(6cos^2(2x) + 2cos^3(2x))/8
This can be further simplified by factoring out a common factor of 2cos^2(2x):
(2cos^2(2x)(3 + cos(2x)))/8
Reducing the fraction, we get:
cos^2(2x)(3 + cos(2x))/4
Now, we can compare the simplified LHS and RHS expressions:
LHS: cos2x(1 - cos^2xsin^2x)
RHS: cos^2(2x)(3 + cos(2x))/4
By comparing the two expressions, we can see that they are not equal to each other. Therefore, the original equation (cos^2x - sin^2x)(1 - cos^2xsin^2x) = (cos^6x - sin^6x) is incorrect.
Hence, the proof reveals that the two sides of the equation are not equal.