Question

Sin^4 x*Cos^2 x-Cos^4 x *Sin^2 x= Cos2x

Answer 1

Sin(x)^4 cos(x)^2 - cos(x)^4 sin(x)^2 = cos(2x)
sin(x)^2 cos(x)^2 [ sin(x)^2 - cos(x)^2 ] = cos(2x)
- sin(x)^2 cos(x)^2 cos(2x) = cos(2x)
cos(2x) [ 1 + sin(x)^2 cos(x)^2 ] = 0

1 + sin(x)^2 cos(x)^2 = 0
(sin(x) cos(x))^2 = -1
Решений нет

cos(2x) = 0
2x = п/2 + пк
x = (п/4)(2k + 1)

Answer 2

Let's prove the given trigonometric identity:

sin^4(x) * cos^2(x) - cos^4(x) * sin^2(x) = cos(2x)

To do this, we'll start by expressing the right-hand side (RHS) of the identity in terms of sin(x) and cos(x) using the double angle formula for cosine:

cos(2x) = cos^2(x) - sin^2(x)

Now, let's manipulate the left-hand side (LHS) of the identity to see if it matches the RHS:

LHS = sin^4(x) * cos^2(x) - cos^4(x) * sin^2(x)

We can factor out common terms from both terms in the LHS:

LHS = (sin^2(x) * cos^2(x)) * sin^2(x) - (cos^2(x) * sin^2(x)) * cos^2(x)

Now, we can use the trigonometric identity sin^2(x) + cos^2(x) = 1:

LHS = (1 - cos^2(x)) * sin^2(x) - (cos^2(x)) * (1 - cos^2(x))

Distribute the terms:

LHS = sin^2(x) - cos^2(x) * sin^2(x) - cos^2(x) + cos^4(x)

Now, use the trigonometric identity 1 - sin^2(x) = cos^2(x):

LHS = (1 - sin^2(x)) - cos^2(x) * sin^2(x) - cos^2(x) + cos^4(x)

LHS = 1 - sin^2(x) - cos^2(x) * sin^2(x) - cos^2(x) + cos^4(x)

Again, use the trigonometric identity sin^2(x) + cos^2(x) = 1:

LHS = 1 - sin^2(x) - cos^2(x) * (1 - cos^2(x)) - cos^2(x) + cos^4(x)

LHS = 1 - sin^2(x) - cos^2(x) + cos^2(x) * cos^2(x) - cos^2(x) + cos^4(x)

Combine like terms:

LHS = 1 - sin^2(x) + cos^2(x) * cos^2(x) + cos^4(x) - cos^2(x)

Use the trigonometric identity sin^2(x) + cos^2(x) = 1 once more:

LHS = 1 - sin^2(x) + cos^4(x) - cos^2(x)

Now, we notice that 1 - sin^2(x) = cos^2(x):

LHS = cos^2(x) + cos^4(x) - cos^2(x)

The cos^2(x) terms cancel out:

LHS = cos^4(x)

Now, we have:

LHS = cos^4(x)

This matches the RHS we derived earlier:

cos(2x) = cos^2(x) - sin^2(x)

Since cos^2(x) - sin^2(x) and cos^4(x) are equivalent, the given identity is proved:

sin^4(x) * cos^2(x) - cos^4(x) * sin^2(x) = cos(2x)