2sin2xcos2x
cos^4x-sin^4x
(sinx+cosx)^2-sin2x
2sinxcos^3x+2sin^3xcosx
To simplify the given expressions, we will use trigonometric identities and properties along with algebraic manipulations.
1) 2sin2xcos2x:
We can use the double-angle identity for sine: sin(2x) = 2sin(x)cos(x).
Similarly, we can use the double-angle identity for cosine: cos(2x) = cos^2(x) - sin^2(x).
Substituting these identities into the given expression, we have:
2sin2xcos2x = 2(2sin(x)cos(x))(cos^2(x) - sin^2(x))
Expanding the expression further:
2sin2xcos2x = 4sin(x)cos(x)(cos^2(x) - sin^2(x))
Distributing the 4sin(x)cos(x) term:
2sin2xcos2x = 4sin(x)cos^3(x) - 4sin^3(x)cos(x)
So, the simplified expression is: 4sin(x)cos^3(x) - 4sin^3(x)cos(x).
2) cos^4x - sin^4x:
We can use the difference of squares identity: a^2 - b^2 = (a + b)(a - b).
Applying this identity to the given expression, we have:
cos^4x - sin^4x = (cos^2x + sin^2x)(cos^2x - sin^2x)
Using the Pythagorean identity: sin^2x + cos^2x = 1, we can simplify the expression:
cos^4x - sin^4x = (1)(cos^2x - sin^2x)
Simplifying further:
cos^4x - sin^4x = cos^2x - sin^2x.
3) (sinx + cosx)^2 - sin2x:
We will first expand the square of the binomial (sinx + cosx)^2:
(sin x + cos x)^2 = sin^2x + 2sinx cosx + cos^2x.
Substituting this expansion into the given expression, we get:
(sin x + cos x)^2 - sin 2x = sin^2x + 2sin x cos x + cos^2x - sin 2x.
Using the double-angle identity for sine: sin 2x = 2sin x cos x,
we can simplify the expression further:
(sin x + cos x)^2 - sin 2x = sin^2x + 2sin x cos x + cos^2x - 2sin x cos x.
Combining like terms:
(sin x + cos x)^2 - sin 2x = sin^2x + cos^2x.
Using the Pythagorean identity again: sin^2x + cos^2x = 1,
we can simplify the expression to:
(sin x + cos x)^2 - sin 2x = 1.
4) 2sinx cos^3x + 2sin^3x cosx:
Factoring out 2sinx cosx from both terms:
2sinx cos^3x + 2sin^3x cosx = 2sinx cosx(cos^2x + sin^2x).
Using the Pythagorean identity once again: cos^2x + sin^2x = 1,
we can simplify the expression to:
2sinx cos^3x + 2sin^3x cosx = 2sinx cosx(1).
Simplifying further by multiplying:
2sinx cos^3x + 2sin^3x cosx = 2sinx cosx.
So the simplified expression is: 2sinx cosx.