Question

2√2*sin(x)+√2sinx*cosx-cosx-2=0

Answer 1

$2\sqrt2\cdot sina+\sqrt2sinx\cdot cosx-cosx-2=0\\\\\sqrt2sinx\cdot (2+cosx)-(cosx+2)=0\\\\(2+cosx)\cdot (\sqrt2sinx-1)=0\\\\a)\; \; 2+cosx=0\; \; \to \; \; \; cosx=-2$

Answer 2

The given equation is a trigonometric equation involving the sine and cosine functions. It's a bit challenging to solve directly because it contains both the sine and cosine functions multiplied together. However, we can still attempt to simplify and solve it.

Let's rewrite the equation:

2√2sin(x) + √2sin(x)*cos(x) - cos(x) - 2 = 0

Notice that there are terms involving both sin(x) and cos(x) multiplied together, which makes it a bit complex. We can use trigonometric identities to simplify the equation and hopefully make it more manageable.

The identity that relates the product of sine and cosine to a sine of a sum is:

sin(A)*cos(B) = (1/2) * [sin(A + B) + sin(A - B)]

Using this identity, we can rewrite the equation:

2√2*sin(x) + (1/2) * √2 * [sin(2x) + sin(0)] - cos(x) - 2 = 0

Simplify the equation further:

2√2*sin(x) + √2 * sin(2x)/2 - cos(x) - 2 = 0

Now, we have the equation in terms of sin(x) and sin(2x). We can use the double angle formula for sine:

sin(2x) = 2 * sin(x) * cos(x)

Replace sin(2x) in the equation:

2√2*sin(x) + √2 * [2 * sin(x) * cos(x)]/2 - cos(x) - 2 = 0

Simplify:

2√2*sin(x) + √2 * sin(x) * cos(x) - cos(x) - 2 = 0

Notice that we've come back to the original equation. Unfortunately, we couldn't simplify it further using these trigonometric identities.

At this point, solving the equation for x analytically might not be straightforward. You might need to use numerical methods or a calculator to approximate the solutions. You could also use software like Mathematica or Python with libraries like SymPy to attempt to find the solutions symbolically or numerically.

If you're looking for specific solutions or more information, please let me know, and I'd be happy to assist you further.