Prove the identity: cos^4(x)-sin^4(x)=1-2sin^2(x).with explanation
im confused on this one cause of the 4th power of the sine. does this mean that we put it in the calculator 4 times? and what exactly does it mean by prove the identity- does it mean like make sure both sides equal each other?
cos ^ 4( x ) - sin ^ 4 ( x ) =
[ cos ^ 2 ( x ) ] ^ 2 - [ sin ^ 2 ( x ) ] ^ 2 =
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Remark:
a ^ 2 - b ^ 2 = ( a + b ) * ( a - b )
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= [ cos ^ 2 ( x ) + sin ^ 2 ( x ) ] *
[ cos ^ 2 ( x ) - sin ^ 2 ( x ) ] =
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Remark:
cos ^ 2 ( x ) + sin ^ 2 ( x ) = 1
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= 1 * [ cos ^ 2 ( x ) - sin ^ 2 ( x ) ] =
cos ^ 2 ( x ) - sin ^ 2 ( x ) =
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Remark:
cos ^ 2 ( x ) = 1 - sin ^ 2 ( x )
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= 1 - sin ^ 2 ( x ) - sin ^ 2 ( x ) =
1 - 2 sin ^ 2 ( x )
To prove the identity cos^4(x) - sin^4(x) = 1 - 2sin^2(x), we need to show that both sides of the equation are equal.
First, let's simplify the left side of the equation using the trigonometric identity for a difference of squares:
cos^4(x) - sin^4(x) = (cos^2(x) + sin^2(x))(cos^2(x) - sin^2(x))
Using the Pythagorean identity sin^2(x) + cos^2(x) = 1, we can substitute that in:
= (1)(cos^2(x) - sin^2(x))
Next, we can use another trigonometric identity, cos^2(x) - sin^2(x) = cos(2x). Substituting this in:
= (1)(cos(2x))
Simplifying further, we have:
= cos(2x)
Now let's simplify the right side of the equation:
1 - 2sin^2(x)
Using the Pythagorean identity sin^2(x) + cos^2(x) = 1, we can rewrite it as:
= 1 - 2(1 - cos^2(x))
= 1 - 2 + 2cos^2(x)
= -1 + 2cos^2(x)
Next, we can use another trigonometric identity, cos^2(x) = (1 + cos(2x))/2:
= -1 + 2((1 + cos(2x))/2)
= -1 + (1 + cos(2x))
= cos(2x)
So, we can see that the left side (cos^4(x) - sin^4(x)) simplifies to cos(2x), and the right side (1 - 2sin^2(x)) also simplifies to cos(2x). Therefore, the two sides of the equation are equal, proving the identity.
Now, regarding your confusion about the 4th power of sine, you don't need to put it in the calculator four times. In this case, cos^4(x) means (cos(x))^4 and sin^4(x) means (sin(x))^4, which is just squaring the respective trigonometric functions twice.
Proving an identity means showing that both sides of the equation are equal. In this case, we have shown that when we simplify both the left and right sides of the equation, they both simplify to cos(2x), proving that the identity holds true.