Question

F(x)=1/(2sin3x) 2(sin2x+cos2x)/cosx-sinx-cos3x+sin3x

Answer 1

sin2x + cos2x = 1

tgx  =  sinxcosx ctgx  =  cosxsinx

tgx ctgx = 1

tg2x + 1  =  1cos2x ctg2x + 1  =  1sin2xФормулы двойного аргумента

sin2x = 2sinx cosx

sin2x  =  2tgx  = 2ctgx  = 21 + tg2x1 + ctg2xtgx + ctgx

cos2x = cos2x - sin2x = 2cos2x - 1 = 1 - 2sin2x

cos2x  =  1 - tg2x  = ctg2x - 1  = ctgx - tgx1 + tg2xctg2x + 1ctgx + tgx tg2x  =  2tgx  = 2ctgx  = 21 - tg2xctg2x - 1ctgx - tgx ctg2x  =  ctg2x - 1  = ctgx - tgx2ctgx2

Answer 2

To simplify the expression, let's start by breaking it down step by step.

Given function: F(x) = 1 / (2sin(3x)) * [2(sin(2x) + cos(2x))] / [cos(x) - sin(x) - cos(3x) + sin(3x)]

Step 1: Simplify the numerator of the fraction. The numerator is 2(sin(2x) + cos(2x)). We can further simplify this expression using the trigonometric identity sin(2x) + cos(2x) = 1.

Numerator = 2(sin(2x) + cos(2x)) = 2(1) = 2

Step 2: Simplify the denominator of the fraction. The denominator is cos(x) - sin(x) - cos(3x) + sin(3x).

Step 3: Expand the trigonometric expressions. We'll expand the trigonometric expressions in the denominator to simplify further.

Denominator = cos(x) - sin(x) - cos(3x) + sin(3x) = cos(x) - sin(x) - (cos(x)cos(2x) - sin(x)sin(2x)) + (sin(x)cos(2x) + cos(x)sin(2x))

Step 4: Combine like terms. Now we can combine like terms in the denominator.

Denominator = cos(x) - sin(x) - cos(x)cos(2x) + sin(x)sin(2x) + sin(x)cos(2x) + cos(x)sin(2x) = cos(x) - cos(x)cos(2x) + sin(x)cos(2x) + sin(x)sin(2x) + sin(x) + cos(x)sin(2x)

Step 5: Rearrange terms. We can rearrange the terms in the denominator for better organization.

Denominator = cos(x) - cos(x)cos(2x) + sin(x)cos(2x) + sin(x)sin(2x) + sin(x) + cos(x)sin(2x) = - cos(x)cos(2x) + sin(x)cos(2x) + cos(x)sin(2x) + sin(x)sin(2x) + cos(x) + sin(x)

Step 6: Simplify the denominator further. We can factor out common terms from the denominator to simplify it.

Denominator = cos(2x)(sin(x) + cos(x)) + sin(2x)(sin(x) + cos(x)) + cos(x) + sin(x) = (sin(x) + cos(x))(cos(2x) + sin(2x)) + cos(x) + sin(x)

Step 7: Simplify the denominator even further. Using the trigonometric identity cos(2x) + sin(2x) = 1, we can simplify the denominator.

Denominator = (sin(x) + cos(x))(cos(2x) + sin(2x)) + cos(x) + sin(x) = (sin(x) + cos(x))(1) + cos(x) + sin(x) = sin(x) + cos(x) + cos(x) + sin(x) = 2sin(x) + 2cos(x)

Step 8: Simplify the expression. Now that we have the simplified numerator and denominator, we can rewrite the expression.

F(x) = 1 / (2sin(3x)) * [2(sin(2x) + cos(2x))] / [cos(x) - sin(x) - cos(3x)