Question

Sin2xcosx-sinxcos2x=-1

Answer 1

$\sin 2x\cos x-\sin x\cos 2x=-1\\ \sin (2x-x)=-1\\ \sin x=-1\\ x=-\frac{\pi}{2}+2\pi k,k \in \mathbb{Z}$

Answer 2

Ответ:

Объяснение:

2sinxcosx×cosx-sinx×(cos^2x-sin^2x)=-1

2sinxcos^2x -sinxcos^2x+sin^3x= -1

sinxcos^2x+sin^3x=-1

Sinx(cos^2x+sin^2x)=-1

sin x=-1

X=pi/2+2pi×n

Answer 3

To solve the equation sin(2x)cos(x) - sin(x)cos(2x) = -1, we can simplify it using trigonometric identities.

Let's start by expanding the trigonometric functions:

sin(2x)cos(x) - sin(x)cos(2x) = -1

Using the double angle formula for sine (sin(2x) = 2sin(x)cos(x)) and double angle formula for cosine (cos(2x) = cos^2(x) - sin^2(x)), we can substitute them into the equation:

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

Next, we can simplify further:

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

Now, let's group the terms:

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

Now, we can factor out sin(x)cos^2(x) from the first two terms:

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

Simplifying the coefficients:

3sin(x)cos^2(x) + sin(x)sin^2(x) = -1

Now, we can rewrite sin^2(x) as 1 - cos^2(x):

3sin(x)cos^2(x) + sin(x)(1 - cos^2(x)) = -1

Expanding the terms:

3sin(x)cos^2(x) + sin(x) - sin(x)cos^2(x) = -1

Combining like terms:

3sin(x)cos^2(x) - sin(x)cos^2(x) + sin(x) = -1

Now, we can factor out sin(x):

sin(x)(3cos^2(x) - cos^2(x) + 1) = -1

Simplifying the coefficients:

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

Finally, we have:

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

Now, we can solve this equation for sin(x) and cos(x) separately. Please note that there may be multiple solutions depending on the range of values for x.