2) Use the sum and difference identites

sin[x + pi/4] + sin[x-pi/4] = -1

sinx cospi/4 + cosxsin pi/4 + sinx cos pi/4 - cosx sin pi/4 = -1

2 sin x cos pi/4 =-1

cos pi/4 = sqr2/2

2sin^x(sqrt2/2) = -1
sin x = -sqrt2

x = 7pi/4 and 5pi/4

Am I correct?

2sin^x(sqrt2/2) = -1

sin x = -1/sqrt2 = - sqrt 2 /2
x = pi + pi/4 = 5 pi/4
x = 2 pi - pi/4 = 7 pi /4

so I agree with you but you seem to have a typo

To confirm if you are correct, let's analyze step by step the solution process.

Starting from the given equation:
sin[x + pi/4] + sin[x - pi/4] = -1

We can use the sum and difference identities for sine to simplify this expression:

sin(x)cos(pi/4) + cos(x)sin(pi/4) + sin(x)cos(pi/4) - cos(x)sin(pi/4) = -1

Simplifying further:

2sin(x)cos(pi/4) = -1

Since cos(pi/4) = sqrt(2)/2, we substitute this value:

2sin(x)(sqrt(2)/2) = -1

sin(x) = -1/(2 * sqrt(2)/2)
sin(x) = -1/sqrt(2)
sin(x) = -sqrt(2)/2

At this point, you wrote "sin x = -sqrt2". However, the correct value for sin(x) is -sqrt(2)/2.

Now, to find the values of x that satisfy this equation, we look for angles whose sine is -sqrt(2)/2. These angles can be found in the third and fourth quadrants.

In the third quadrant (where sin is negative), the angle x is:

x = 7pi/4

In the fourth quadrant (also where sin is negative), the angle x is:

x = 5pi/4

Therefore, the correct solutions for the equation are x = 7pi/4 and x = 5pi/4.

In conclusion, you are correct in your solution.