2sin2xcos2x
cos^4x-sin^4x
(sinx+cosx)^2-sin2x
2sinxcos^3x+2sin^3xcosx
To simplify these expressions, we'll use some trigonometric identities. Here's how to approach each one:
1. 2sin2xcos2x:
- Start by using the double angle identity for sine: sin(2x) = 2sin(x)cos(x).
- Substitute this identity into the expression: 2(2sin(x)cos(x))cos(2x).
- Simplify: 4sin(x)cos^2(x)cos(2x).
- Next, use the double angle identity for cosine: cos(2x) = cos^2(x) - sin^2(x).
- Substitute this identity into the expression: 4sin(x)cos^2(x)(cos^2(x) - sin^2(x)).
- Simplify further: 4sin(x)cos^4(x) - 4sin(x)cos^2(x)sin^2(x).
2. cos^4x - sin^4x:
- Notice that this expression consists of the difference of two squares.
- Recall the identity: a^2 - b^2 = (a + b)(a - b).
- Apply the identity to the given expression: (cos^2x + sin^2x)(cos^2x - sin^2x).
- Simplify: 1(cos^2x - sin^2x) = cos^2x - sin^2x.
3. (sinx + cosx)^2 - sin2x:
- Start by expanding the square: (sinx + cosx)(sinx + cosx) - sin2x.
- Multiply each term in the first parentheses with each term in the second parentheses: sin^2x + sinxcosx + cosxsinx + cos^2x - sin2x.
- Combine like terms: sin^2x + 2sinxcosx + cos^2x - sin2x.
- Simplify further: sin^2x + cos^2x + 2sinxcosx - sin2x.
- Recall the Pythagorean identity sin^2x + cos^2x = 1: 1 + 2sinxcosx - sin2x.
4. 2sinxcos^3x + 2sin^3xcosx:
- Notice that this expression consists of two terms that are similar except for the order of the sine and cosine.
- Factor out sinx in the first term: sinx(2cos^3x + 2sin^2xcosx).
- Factor out cosx in the second term: cosx(2sin^3x + 2sinxcos^2x).
- Simplify further: 2sinx(cos^3x + sin^2xcosx) + 2cosx(sin^3x + sinxcos^2x).
- Apply the identity: sin^2x = 1 - cos^2x.
- Substitute this identity into the expression: 2sinx(cos^3x + (1 - cos^2x)cosx) + 2cosx(sin^3x + sinxcos^2x).
- Simplify: 2sinx(cos^3x + cosx - cos^3x) + 2cosx(sin^3x + sinxcos^2x).
- Further simplify: 2sinx(cosx) + 2cosx(sin^3x + sinxcos^2x).
- Finally, factor out 2sinx and 2cosx respectively: 2sinxcosx + 2cosx(sin^3x + sinxcos^2x).