Simplify the trigonometric function
sin^4x-cos^4x
cos^2â-sin^2â=1+2cosâ
(1+cot^2x )(cos^2x )=cot^2x
cot^2t/csct =(1-sin^2t)/sint (Work on both sides!)
sinècscè- sin^2è=cos^2è
sin^4x-cos^4x = (sin^2 x+cos^2 x)(sin^2 x -cos^2 x
= 1 (sin^2 x - cos^2 x)
= 2 sin^2 x - 1
To simplify the trigonometric function sin^4(x) - cos^4(x), we can first use the identity:
cos^2(α) - sin^2(α) = 1
Let's rewrite the expression using this identity:
sin^4(x) - cos^4(x) = (sin^2(x))^2 - (cos^2(x))^2
Now, we can substitute sin^2(x) with (1 - cos^2(x)) using the identity:
1 - cos^2(α) = sin^2(α)
(sin^2(x))^2 - (cos^2(x))^2 = ((1 - cos^2(x)))^2 - (cos^2(x))^2
Expanding the square and simplifying, we get:
(1 - 2cos^2(x) + cos^4(x)) - cos^2(x)^2 = 1 - 2cos^2(x) + cos^4(x) - cos^4(x)
The terms cos^4(x) cancel out, leaving us with:
1 - 2cos^2(x)
Therefore, the simplified form of the trigonometric function sin^4(x) - cos^4(x) is 1 - 2cos^2(x).